Data.Colour.Matrix:inverse from colour-2.3.3, B

Percentage Accurate: 91.2% → 97.1%

Time: 7.9s

Alternatives: 10

Speedup: 0.7×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (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 * t)) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (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 * t)) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+300} \lor \neg \left(t_1 \leq 4 \cdot 10^{+260}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

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 (- (* x y) (* z t))))
   (if (or (<= t_1 -2e+300) (not (<= t_1 4e+260)))
     (fma x (/ y a) (/ (- z) (/ a t)))
     (/ t_1 a))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -2e+300) || !(t_1 <= 4e+260)) {
		tmp = fma(x, (y / a), (-z / (a / t)));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -2e+300) || !(t_1 <= 4e+260))
		tmp = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)));
	else
		tmp = Float64(t_1 / a);
	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[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+300], N[Not[LessEqual[t$95$1, 4e+260]], $MachinePrecision]], N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000001e300 or 4.00000000000000026e260 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 69.8%

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

      1. div-sub 65.5%

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

      2. *-un-lft-identity 65.5%

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

      3. times-frac 80.4%

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

      4. fma-neg 83.3%

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

      5. associate-/l* 96.9%

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

   3. Applied egg-rr 96.9%

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

   if -2.0000000000000001e300 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.00000000000000026e260

   1. Initial program 97.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+300} \lor \neg \left(x \cdot y - z \cdot t \leq 4 \cdot 10^{+260}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 2: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \end{array} \]

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 (- (* x y) (* z t))))
   (if (<= t_1 -2e+300)
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (if (<= t_1 5e+307) (/ t_1 a) (- (/ x (/ a y)) (/ z (/ a t)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else if (t_1 <= 5e+307) {
		tmp = t_1 / a;
	} else {
		tmp = (x / (a / y)) - (z / (a / t));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -2e+300)
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	elseif (t_1 <= 5e+307)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	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[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], N[(t$95$1 / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000001e300

   1. Initial program 69.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 66.3%

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

      1. fma-def 66.3%

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

      2. associate-/l* 75.2%

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

      3. associate-/l* 93.3%

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

   4. Simplified 93.3%

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

   if -2.0000000000000001e300 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5e307

   1. Initial program 97.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 5e307 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 64.5%

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

      1. div-sub 58.2%

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

      2. associate-/l* 72.8%

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

      3. associate-/l* 90.5%

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

   3. Applied egg-rr 90.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 3: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

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 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+307)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ t_1 a))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+307)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e+307)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e+307):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+307))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e+307)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = t_1 / a;
	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_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+307]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 5e307 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 65.9%

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

      1. div-sub 61.0%

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

      2. associate-/l* 77.8%

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

      3. associate-/l* 93.2%

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

   3. Applied egg-rr 93.2%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5e307

   1. Initial program 97.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 4: 71.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1.85 \cdot 10^{-20}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e+50)
   (/ y (/ a x))
   (if (<= (* x y) -1.85e-20)
     (* (- t) (/ z a))
     (if (<= (* x y) -5e-119)
       (* y (/ x a))
       (if (<= (* x y) 0.0)
         (/ (- t) (/ a z))
         (if (<= (* x y) 4e-89) (/ (- (* z t)) a) (* x (/ y a))))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e+50) {
		tmp = y / (a / x);
	} else if ((x * y) <= -1.85e-20) {
		tmp = -t * (z / a);
	} else if ((x * y) <= -5e-119) {
		tmp = y * (x / a);
	} else if ((x * y) <= 0.0) {
		tmp = -t / (a / z);
	} else if ((x * y) <= 4e-89) {
		tmp = -(z * t) / a;
	} else {
		tmp = x * (y / a);
	}
	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 ((x * y) <= (-2d+50)) then
        tmp = y / (a / x)
    else if ((x * y) <= (-1.85d-20)) then
        tmp = -t * (z / a)
    else if ((x * y) <= (-5d-119)) then
        tmp = y * (x / a)
    else if ((x * y) <= 0.0d0) then
        tmp = -t / (a / z)
    else if ((x * y) <= 4d-89) then
        tmp = -(z * t) / a
    else
        tmp = x * (y / a)
    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 ((x * y) <= -2e+50) {
		tmp = y / (a / x);
	} else if ((x * y) <= -1.85e-20) {
		tmp = -t * (z / a);
	} else if ((x * y) <= -5e-119) {
		tmp = y * (x / a);
	} else if ((x * y) <= 0.0) {
		tmp = -t / (a / z);
	} else if ((x * y) <= 4e-89) {
		tmp = -(z * t) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e+50:
		tmp = y / (a / x)
	elif (x * y) <= -1.85e-20:
		tmp = -t * (z / a)
	elif (x * y) <= -5e-119:
		tmp = y * (x / a)
	elif (x * y) <= 0.0:
		tmp = -t / (a / z)
	elif (x * y) <= 4e-89:
		tmp = -(z * t) / a
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+50)
		tmp = Float64(y / Float64(a / x));
	elseif (Float64(x * y) <= -1.85e-20)
		tmp = Float64(Float64(-t) * Float64(z / a));
	elseif (Float64(x * y) <= -5e-119)
		tmp = Float64(y * Float64(x / a));
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(Float64(-t) / Float64(a / z));
	elseif (Float64(x * y) <= 4e-89)
		tmp = Float64(Float64(-Float64(z * t)) / a);
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e+50)
		tmp = y / (a / x);
	elseif ((x * y) <= -1.85e-20)
		tmp = -t * (z / a);
	elseif ((x * y) <= -5e-119)
		tmp = y * (x / a);
	elseif ((x * y) <= 0.0)
		tmp = -t / (a / z);
	elseif ((x * y) <= 4e-89)
		tmp = -(z * t) / a;
	else
		tmp = x * (y / a);
	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[N[(x * y), $MachinePrecision], -2e+50], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1.85e-20], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-119], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e-89], N[((-N[(z * t), $MachinePrecision]) / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (*.f64 x y) < -2.0000000000000002e50

   1. Initial program 82.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 76.8%

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

      1. associate-/l* 84.3%

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

   4. Simplified 84.3%

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

   if -2.0000000000000002e50 < (*.f64 x y) < -1.85e-20

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 65.8%

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

      2. mul-1-neg 65.8%

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

      3. distribute-rgt-neg-out 65.8%

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

      4. *-commutative 65.8%

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

      5. associate-/l* 65.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 59.1%

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

   4. Simplified 59.1%

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

   if -1.85e-20 < (*.f64 x y) < -4.99999999999999993e-119

   1. Initial program 93.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 65.3%

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

      1. associate-*r/ 72.0%

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

   4. Simplified 72.0%

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

   if -4.99999999999999993e-119 < (*.f64 x y) < 0.0

   1. Initial program 85.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 78.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 78.9%

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

      2. mul-1-neg 78.9%

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

      3. distribute-rgt-neg-out 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-/l* 86.4%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 85.9%

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

   4. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 78.9%

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

      2. *-commutative 78.9%

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

      3. add-sqr-sqrt 46.9%

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

      4. sqrt-unprod 39.6%

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

      5. sqr-neg 39.6%

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

      6. sqrt-unprod 3.1%

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

      7. add-sqr-sqrt 14.0%

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

      8. associate-/l* 14.0%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Applied egg-rr 14.0%

      \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 13.9%

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

   8. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{\frac{t}{a} \cdot z} \]
   9. Step-by-step derivation

      1. div-inv 13.9%

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

      2. add-sqr-sqrt 7.9%

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

      3. sqrt-unprod 50.1%

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

      4. sqr-neg 50.1%

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

      5. sqrt-unprod 56.1%

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

      6. add-sqr-sqrt 85.8%

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

      7. associate-*r* 85.9%

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

      8. associate-/r/ 85.7%

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

      9. un-div-inv 85.9%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      10. frac-2neg 85.9%

          \[\leadsto \color{blue}{\frac{-t}{-\frac{-a}{z}}} \]

      11. distribute-frac-neg 85.9%

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

      12. remove-double-neg 85.9%

          \[\leadsto \frac{-t}{\color{blue}{\frac{a}{z}}} \]

   10. Applied egg-rr 85.9%

       \[\leadsto \color{blue}{\frac{-t}{\frac{a}{z}}} \]

   if 0.0 < (*.f64 x y) < 4.00000000000000015e-89

   1. Initial program 97.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 85.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 85.1%

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

      2. associate-*r* 85.1%

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

      3. neg-mul-1 85.1%

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

   4. Simplified 85.1%

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

   if 4.00000000000000015e-89 < (*.f64 x y)

   1. Initial program 89.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 68.2%

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

      1. associate-*r/ 69.7%

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

   4. Simplified 69.7%

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

      1. associate-*r/ 68.2%

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

      2. clear-num 68.1%

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

      3. *-commutative 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. *-commutative 68.1%

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

      2. clear-num 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r/ 63.4%

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

      5. *-commutative 63.4%

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

   8. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \cdot y \leq -1.85 \cdot 10^{-20}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5: 65.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-38} \lor \neg \left(z \leq 5.6 \cdot 10^{-130}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

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 (* z (- (/ t a)))))
   (if (<= z -4.3e+57)
     t_1
     (if (<= z -3.15e-21)
       (* y (/ x a))
       (if (or (<= z -4e-38) (not (<= z 5.6e-130))) t_1 (/ (* x y) a))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = z * -(t / a);
	double tmp;
	if (z <= -4.3e+57) {
		tmp = t_1;
	} else if (z <= -3.15e-21) {
		tmp = y * (x / a);
	} else if ((z <= -4e-38) || !(z <= 5.6e-130)) {
		tmp = t_1;
	} else {
		tmp = (x * y) / a;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = z * -(t / a)
    if (z <= (-4.3d+57)) then
        tmp = t_1
    else if (z <= (-3.15d-21)) then
        tmp = y * (x / a)
    else if ((z <= (-4d-38)) .or. (.not. (z <= 5.6d-130))) then
        tmp = t_1
    else
        tmp = (x * y) / a
    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 t_1 = z * -(t / a);
	double tmp;
	if (z <= -4.3e+57) {
		tmp = t_1;
	} else if (z <= -3.15e-21) {
		tmp = y * (x / a);
	} else if ((z <= -4e-38) || !(z <= 5.6e-130)) {
		tmp = t_1;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = z * -(t / a)
	tmp = 0
	if z <= -4.3e+57:
		tmp = t_1
	elif z <= -3.15e-21:
		tmp = y * (x / a)
	elif (z <= -4e-38) or not (z <= 5.6e-130):
		tmp = t_1
	else:
		tmp = (x * y) / a
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(-Float64(t / a)))
	tmp = 0.0
	if (z <= -4.3e+57)
		tmp = t_1;
	elseif (z <= -3.15e-21)
		tmp = Float64(y * Float64(x / a));
	elseif ((z <= -4e-38) || !(z <= 5.6e-130))
		tmp = t_1;
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = z * -(t / a);
	tmp = 0.0;
	if (z <= -4.3e+57)
		tmp = t_1;
	elseif (z <= -3.15e-21)
		tmp = y * (x / a);
	elseif ((z <= -4e-38) || ~((z <= 5.6e-130)))
		tmp = t_1;
	else
		tmp = (x * y) / a;
	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_] := Block[{t$95$1 = N[(z * (-N[(t / a), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[z, -4.3e+57], t$95$1, If[LessEqual[z, -3.15e-21], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -4e-38], N[Not[LessEqual[z, 5.6e-130]], $MachinePrecision]], t$95$1, N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.30000000000000033e57 or -3.15e-21 < z < -3.9999999999999998e-38 or 5.60000000000000032e-130 < z

   1. Initial program 87.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 58.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 58.2%

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

      2. mul-1-neg 58.2%

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

      3. distribute-rgt-neg-out 58.2%

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

      4. *-commutative 58.2%

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

      5. associate-/l* 62.6%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 63.6%

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

   4. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 58.2%

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

      2. *-commutative 58.2%

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

      3. add-sqr-sqrt 24.0%

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

      4. sqrt-unprod 23.3%

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

      5. sqr-neg 23.3%

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

      6. sqrt-unprod 2.7%

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

      7. add-sqr-sqrt 3.8%

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

      8. associate-/l* 6.5%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Applied egg-rr 6.5%

      \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 3.7%

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

      2. sqrt-unprod 27.5%

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

      3. sqr-neg 27.5%

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

      4. sqrt-unprod 30.8%

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

      5. add-sqr-sqrt 63.5%

         \[\leadsto \frac{\color{blue}{-t}}{\frac{a}{z}} \]

      6. distribute-frac-neg 63.5%

         \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]

      7. associate-/r/ 62.2%

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

      8. distribute-rgt-neg-in 62.2%

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

   8. Applied egg-rr 62.2%

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

   if -4.30000000000000033e57 < z < -3.15e-21

   1. Initial program 92.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 63.6%

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

      1. associate-*r/ 55.7%

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

   4. Simplified 55.7%

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

   if -3.9999999999999998e-38 < z < 5.60000000000000032e-130

   1. Initial program 93.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 66.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+57}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-38} \lor \neg \left(z \leq 5.6 \cdot 10^{-130}\right):\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 6: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \left(-t\right) \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.45 \cdot 10^{-17}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (* (- t) (/ z a))))
   (if (<= z -4.2e+57)
     t_1
     (if (<= z -4.45e-17)
       (* y (/ x a))
       (if (<= z -1.2e-40)
         (* z (- (/ t a)))
         (if (<= z 1.3e-130) (/ (* x y) a) t_1))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * (z / a);
	double tmp;
	if (z <= -4.2e+57) {
		tmp = t_1;
	} else if (z <= -4.45e-17) {
		tmp = y * (x / a);
	} else if (z <= -1.2e-40) {
		tmp = z * -(t / a);
	} else if (z <= 1.3e-130) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -t * (z / a)
    if (z <= (-4.2d+57)) then
        tmp = t_1
    else if (z <= (-4.45d-17)) then
        tmp = y * (x / a)
    else if (z <= (-1.2d-40)) then
        tmp = z * -(t / a)
    else if (z <= 1.3d-130) then
        tmp = (x * y) / a
    else
        tmp = t_1
    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 t_1 = -t * (z / a);
	double tmp;
	if (z <= -4.2e+57) {
		tmp = t_1;
	} else if (z <= -4.45e-17) {
		tmp = y * (x / a);
	} else if (z <= -1.2e-40) {
		tmp = z * -(t / a);
	} else if (z <= 1.3e-130) {
		tmp = (x * y) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = -t * (z / a)
	tmp = 0
	if z <= -4.2e+57:
		tmp = t_1
	elif z <= -4.45e-17:
		tmp = y * (x / a)
	elif z <= -1.2e-40:
		tmp = z * -(t / a)
	elif z <= 1.3e-130:
		tmp = (x * y) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) * Float64(z / a))
	tmp = 0.0
	if (z <= -4.2e+57)
		tmp = t_1;
	elseif (z <= -4.45e-17)
		tmp = Float64(y * Float64(x / a));
	elseif (z <= -1.2e-40)
		tmp = Float64(z * Float64(-Float64(t / a)));
	elseif (z <= 1.3e-130)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = -t * (z / a);
	tmp = 0.0;
	if (z <= -4.2e+57)
		tmp = t_1;
	elseif (z <= -4.45e-17)
		tmp = y * (x / a);
	elseif (z <= -1.2e-40)
		tmp = z * -(t / a);
	elseif (z <= 1.3e-130)
		tmp = (x * y) / a;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e+57], t$95$1, If[LessEqual[z, -4.45e-17], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.2e-40], N[(z * (-N[(t / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 1.3e-130], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.19999999999999982e57 or 1.3e-130 < z

   1. Initial program 87.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 58.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 58.1%

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

      2. mul-1-neg 58.1%

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

      3. distribute-rgt-neg-out 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-/l* 62.5%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 63.5%

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

   4. Simplified 63.5%

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

   if -4.19999999999999982e57 < z < -4.4500000000000002e-17

   1. Initial program 92.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 63.6%

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

      1. associate-*r/ 55.7%

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

   4. Simplified 55.7%

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

   if -4.4500000000000002e-17 < z < -1.19999999999999996e-40

   1. Initial program 99.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.5%

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

      2. mul-1-neg 70.5%

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

      3. distribute-rgt-neg-out 70.5%

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

      4. *-commutative 70.5%

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

      5. associate-/l* 70.5%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 70.5%

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

   4. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 70.5%

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

      2. *-commutative 70.5%

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

      3. add-sqr-sqrt 70.5%

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

      4. sqrt-unprod 70.5%

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

      5. sqr-neg 70.5%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.9%

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

      8. associate-/l* 0.9%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Applied egg-rr 0.9%

      \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 0.4%

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

      3. sqr-neg 0.4%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 70.5%

         \[\leadsto \frac{\color{blue}{-t}}{\frac{a}{z}} \]

      6. distribute-frac-neg 70.5%

         \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]

      7. associate-/r/ 70.5%

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

      8. distribute-rgt-neg-in 70.5%

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

   8. Applied egg-rr 70.5%

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

   if -1.19999999999999996e-40 < z < 1.3e-130

   1. Initial program 93.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 66.5%

      \[\leadsto \color{blue}{\frac{y \cdot x}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+57}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq -4.45 \cdot 10^{-17}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]

Alternative 7: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+57}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-37}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \end{array} \]

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 -3e+57)
   (/ (- t) (/ a z))
   (if (<= z -3.7e-14)
     (* y (/ x a))
     (if (<= z -1.2e-37)
       (* z (- (/ t a)))
       (if (<= z 6.8e-130) (/ (* x y) a) (* (- t) (/ z a)))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e+57) {
		tmp = -t / (a / z);
	} else if (z <= -3.7e-14) {
		tmp = y * (x / a);
	} else if (z <= -1.2e-37) {
		tmp = z * -(t / a);
	} else if (z <= 6.8e-130) {
		tmp = (x * y) / a;
	} else {
		tmp = -t * (z / a);
	}
	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 <= (-3d+57)) then
        tmp = -t / (a / z)
    else if (z <= (-3.7d-14)) then
        tmp = y * (x / a)
    else if (z <= (-1.2d-37)) then
        tmp = z * -(t / a)
    else if (z <= 6.8d-130) then
        tmp = (x * y) / a
    else
        tmp = -t * (z / a)
    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 <= -3e+57) {
		tmp = -t / (a / z);
	} else if (z <= -3.7e-14) {
		tmp = y * (x / a);
	} else if (z <= -1.2e-37) {
		tmp = z * -(t / a);
	} else if (z <= 6.8e-130) {
		tmp = (x * y) / a;
	} else {
		tmp = -t * (z / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if z <= -3e+57:
		tmp = -t / (a / z)
	elif z <= -3.7e-14:
		tmp = y * (x / a)
	elif z <= -1.2e-37:
		tmp = z * -(t / a)
	elif z <= 6.8e-130:
		tmp = (x * y) / a
	else:
		tmp = -t * (z / a)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3e+57)
		tmp = Float64(Float64(-t) / Float64(a / z));
	elseif (z <= -3.7e-14)
		tmp = Float64(y * Float64(x / a));
	elseif (z <= -1.2e-37)
		tmp = Float64(z * Float64(-Float64(t / a)));
	elseif (z <= 6.8e-130)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(Float64(-t) * Float64(z / a));
	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 <= -3e+57)
		tmp = -t / (a / z);
	elseif (z <= -3.7e-14)
		tmp = y * (x / a);
	elseif (z <= -1.2e-37)
		tmp = z * -(t / a);
	elseif (z <= 6.8e-130)
		tmp = (x * y) / a;
	else
		tmp = -t * (z / a);
	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, -3e+57], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.7e-14], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.2e-37], N[(z * (-N[(t / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 6.8e-130], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3e57

   1. Initial program 82.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 66.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 66.4%

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

      2. mul-1-neg 66.4%

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

      3. distribute-rgt-neg-out 66.4%

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

      4. *-commutative 66.4%

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

      5. associate-/l* 77.7%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 74.1%

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

   4. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 66.4%

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

      2. *-commutative 66.4%

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

      3. add-sqr-sqrt 66.3%

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

      4. sqrt-unprod 52.7%

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

      5. sqr-neg 52.7%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 3.2%

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

      8. associate-/l* 9.0%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Applied egg-rr 9.0%

      \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 3.3%

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

   8. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{\frac{t}{a} \cdot z} \]
   9. Step-by-step derivation

      1. div-inv 3.3%

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

      2. add-sqr-sqrt 2.7%

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

      3. sqrt-unprod 38.0%

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

      4. sqr-neg 38.0%

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

      5. sqrt-unprod 44.7%

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

      6. add-sqr-sqrt 77.7%

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

      7. associate-*r* 74.0%

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

      8. associate-/r/ 74.0%

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

      9. un-div-inv 74.0%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      10. frac-2neg 74.0%

          \[\leadsto \color{blue}{\frac{-t}{-\frac{-a}{z}}} \]

      11. distribute-frac-neg 74.0%

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

      12. remove-double-neg 74.0%

          \[\leadsto \frac{-t}{\color{blue}{\frac{a}{z}}} \]

   10. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{\frac{-t}{\frac{a}{z}}} \]

   if -3e57 < z < -3.70000000000000001e-14

   1. Initial program 92.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 63.6%

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

      1. associate-*r/ 55.7%

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

   4. Simplified 55.7%

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

   if -3.70000000000000001e-14 < z < -1.19999999999999995e-37

   1. Initial program 99.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.5%

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

      2. mul-1-neg 70.5%

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

      3. distribute-rgt-neg-out 70.5%

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

      4. *-commutative 70.5%

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

      5. associate-/l* 70.5%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 70.5%

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

   4. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 70.5%

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

      2. *-commutative 70.5%

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

      3. add-sqr-sqrt 70.5%

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

      4. sqrt-unprod 70.5%

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

      5. sqr-neg 70.5%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.9%

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

      8. associate-/l* 0.9%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Applied egg-rr 0.9%

      \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 0.4%

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

      3. sqr-neg 0.4%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 70.5%

         \[\leadsto \frac{\color{blue}{-t}}{\frac{a}{z}} \]

      6. distribute-frac-neg 70.5%

         \[\leadsto \color{blue}{-\frac{t}{\frac{a}{z}}} \]

      7. associate-/r/ 70.5%

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

      8. distribute-rgt-neg-in 70.5%

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

   8. Applied egg-rr 70.5%

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

   if -1.19999999999999995e-37 < z < 6.8000000000000001e-130

   1. Initial program 93.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 66.5%

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

   if 6.8000000000000001e-130 < z

   1. Initial program 90.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 53.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 53.5%

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

      2. mul-1-neg 53.5%

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

      3. distribute-rgt-neg-out 53.5%

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

      4. *-commutative 53.5%

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

      5. associate-/l* 54.1%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 57.7%

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

   4. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+57}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-37}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]

Alternative 8: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \end{array} \]

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

C

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

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = -t / (a / z);
	} else {
		tmp = ((x * y) - (z * t)) / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = -t / (a / z)
	else:
		tmp = ((x * y) - (z * t)) / a
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(-t) / Float64(a / z));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	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 * t) <= -Inf)
		tmp = -t / (a / z);
	else
		tmp = ((x * y) - (z * t)) / a;
	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[N[(z * t), $MachinePrecision], (-Infinity)], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 58.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 64.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

      3. distribute-rgt-neg-out 64.5%

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

      4. *-commutative 64.5%

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

      5. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]

      6. associate-/r/ 99.8%

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

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} \]
   5. Step-by-step derivation

      1. associate-*l/ 64.5%

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

      2. *-commutative 64.5%

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

      3. add-sqr-sqrt 39.0%

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

      4. sqrt-unprod 39.0%

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

      5. sqr-neg 39.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.1%

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

      8. associate-/l* 0.2%

         \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 0.2%

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

   8. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\frac{t}{a} \cdot z} \]
   9. Step-by-step derivation

      1. div-inv 0.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 44.2%

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

      4. sqr-neg 44.2%

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

      5. sqrt-unprod 62.1%

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

      6. add-sqr-sqrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. associate-/r/ 99.6%

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

      9. un-div-inv 99.9%

         \[\leadsto \color{blue}{\frac{t}{\frac{-a}{z}}} \]

      10. frac-2neg 99.9%

          \[\leadsto \color{blue}{\frac{-t}{-\frac{-a}{z}}} \]

      11. distribute-frac-neg 99.9%

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

      12. remove-double-neg 99.9%

          \[\leadsto \frac{-t}{\color{blue}{\frac{a}{z}}} \]

   10. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\frac{-t}{\frac{a}{z}}} \]

   if -inf.0 < (*.f64 z t)

   1. Initial program 92.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 9: 51.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= y 8e-145) (* y (/ x a)) (* x (/ y a))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 8e-145) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	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 (y <= 8d-145) then
        tmp = y * (x / a)
    else
        tmp = x * (y / a)
    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 (y <= 8e-145) {
		tmp = y * (x / a);
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if y <= 8e-145:
		tmp = y * (x / a)
	else:
		tmp = x * (y / a)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 8e-145)
		tmp = Float64(y * Float64(x / a));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 8e-145)
		tmp = y * (x / a);
	else
		tmp = x * (y / a);
	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[y, 8e-145], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999932e-145

   1. Initial program 89.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 46.8%

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

      1. associate-*r/ 48.0%

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

   4. Simplified 48.0%

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

   if 7.99999999999999932e-145 < y

   1. Initial program 91.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 57.1%

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

      1. associate-*r/ 59.3%

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

   4. Simplified 59.3%

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

      1. associate-*r/ 57.1%

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

      2. clear-num 57.0%

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

      3. *-commutative 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. *-commutative 57.0%

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

      2. clear-num 57.1%

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

      3. *-commutative 57.1%

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

      4. associate-*r/ 57.0%

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

      5. *-commutative 57.0%

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

   8. Applied egg-rr 57.0%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10: 51.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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 = y * (x / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around inf 50.7%

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

   1. associate-*r/ 52.3%

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

4. Simplified 52.3%

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

5. Final simplification 52.3%

   \[\leadsto y \cdot \frac{x}{a} \]

Developer target: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023182 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

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