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

Percentage Accurate: 91.2% → 93.3%

Time: 7.1s

Alternatives: 8

Speedup: 0.7×

• 28.0% 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: 93.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.8e+34)
   (- (* (/ y (pow (cbrt a) 2.0)) (/ x (cbrt a))) (* (/ t a) z))
   (/ (- (* y x) (* t z)) a)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.8e+34) {
		tmp = ((y / pow(cbrt(a), 2.0)) * (x / cbrt(a))) - ((t / a) * z);
	} else {
		tmp = ((y * x) - (t * z)) / a;
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.8e+34) {
		tmp = ((y / Math.pow(Math.cbrt(a), 2.0)) * (x / Math.cbrt(a))) - ((t / a) * z);
	} else {
		tmp = ((y * x) - (t * z)) / a;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -6.8e+34)
		tmp = Float64(Float64(Float64(y / (cbrt(a) ^ 2.0)) * Float64(x / cbrt(a))) - Float64(Float64(t / a) * z));
	else
		tmp = Float64(Float64(Float64(y * x) - Float64(t * z)) / 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_] := If[LessEqual[a, -6.8e+34], N[(N[(N[(y / N[Power[N[Power[a, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(x / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.7999999999999999e34

   1. Initial program 79.9%

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

      1. div-sub 79.9%

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

      2. *-commutative 79.9%

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

      3. add-cube-cbrt 79.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      4. times-frac 85.1%

         \[\leadsto \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{x}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 85.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{x}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right)} \]

      6. pow2 85.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{x}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. associate-/l* 94.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{x}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{x}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 94.4%

         \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/l* 85.1%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \color{blue}{\frac{z \cdot t}{a}} \]

      3. *-commutative 85.1%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{\color{blue}{t \cdot z}}{a} \]

      4. associate-/l* 94.3%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \color{blue}{\frac{t}{\frac{a}{z}}} \]

      5. associate-/r/ 94.3%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \color{blue}{\frac{t}{a} \cdot z} \]

   5. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{t}{a} \cdot z} \]

   if -6.7999999999999999e34 < a

   1. Initial program 97.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \end{array} \]

Alternative 2: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := \frac{t}{a} \cdot \left(-z\right)\\ t_2 := x \cdot \frac{y}{a}\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \cdot x \leq 0.05:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;y \cdot x \leq 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

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 a) (- z))) (t_2 (* x (/ y a))))
   (if (<= (* y x) -1e+47)
     t_2
     (if (<= (* y x) 5e-34)
       t_1
       (if (<= (* y x) 0.05) (/ (* y x) a) (if (<= (* y x) 1e+53) t_1 t_2))))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (t / a) * -z;
	double t_2 = x * (y / a);
	double tmp;
	if ((y * x) <= -1e+47) {
		tmp = t_2;
	} else if ((y * x) <= 5e-34) {
		tmp = t_1;
	} else if ((y * x) <= 0.05) {
		tmp = (y * x) / a;
	} else if ((y * x) <= 1e+53) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (t / a) * -z
    t_2 = x * (y / a)
    if ((y * x) <= (-1d+47)) then
        tmp = t_2
    else if ((y * x) <= 5d-34) then
        tmp = t_1
    else if ((y * x) <= 0.05d0) then
        tmp = (y * x) / a
    else if ((y * x) <= 1d+53) then
        tmp = t_1
    else
        tmp = t_2
    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 / a) * -z;
	double t_2 = x * (y / a);
	double tmp;
	if ((y * x) <= -1e+47) {
		tmp = t_2;
	} else if ((y * x) <= 5e-34) {
		tmp = t_1;
	} else if ((y * x) <= 0.05) {
		tmp = (y * x) / a;
	} else if ((y * x) <= 1e+53) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	t_1 = (t / a) * -z
	t_2 = x * (y / a)
	tmp = 0
	if (y * x) <= -1e+47:
		tmp = t_2
	elif (y * x) <= 5e-34:
		tmp = t_1
	elif (y * x) <= 0.05:
		tmp = (y * x) / a
	elif (y * x) <= 1e+53:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(t / a) * Float64(-z))
	t_2 = Float64(x * Float64(y / a))
	tmp = 0.0
	if (Float64(y * x) <= -1e+47)
		tmp = t_2;
	elseif (Float64(y * x) <= 5e-34)
		tmp = t_1;
	elseif (Float64(y * x) <= 0.05)
		tmp = Float64(Float64(y * x) / a);
	elseif (Float64(y * x) <= 1e+53)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (t / a) * -z;
	t_2 = x * (y / a);
	tmp = 0.0;
	if ((y * x) <= -1e+47)
		tmp = t_2;
	elseif ((y * x) <= 5e-34)
		tmp = t_1;
	elseif ((y * x) <= 0.05)
		tmp = (y * x) / a;
	elseif ((y * x) <= 1e+53)
		tmp = t_1;
	else
		tmp = t_2;
	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[(t / a), $MachinePrecision] * (-z)), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e+47], t$95$2, If[LessEqual[N[(y * x), $MachinePrecision], 5e-34], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 0.05], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e+53], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e47 or 9.9999999999999999e52 < (*.f64 x y)

   1. Initial program 88.7%

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

   2. Taylor expanded in x around inf 78.1%

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

      1. associate-*r/ 83.0%

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

   4. Simplified 83.0%

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

   if -1e47 < (*.f64 x y) < 5.0000000000000003e-34 or 0.050000000000000003 < (*.f64 x y) < 9.9999999999999999e52

   1. Initial program 94.8%

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

      1. div-sub 94.8%

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

      2. *-commutative 94.8%

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

      3. add-cube-cbrt 94.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      4. times-frac 91.7%

         \[\leadsto \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{x}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 91.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{x}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right)} \]

      6. pow2 91.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{x}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. associate-/l* 90.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{x}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{x}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 90.4%

         \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/l* 91.7%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \color{blue}{\frac{z \cdot t}{a}} \]

      3. *-commutative 91.7%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{\color{blue}{t \cdot z}}{a} \]

      4. associate-/l* 91.8%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \color{blue}{\frac{t}{\frac{a}{z}}} \]

      5. associate-/r/ 91.1%

         \[\leadsto \frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \color{blue}{\frac{t}{a} \cdot z} \]

   5. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{x}{\sqrt[3]{a}} - \frac{t}{a} \cdot z} \]

   6. Taylor expanded in y around 0 74.5%

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

      1. neg-mul-1 74.5%

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

      2. associate-*l/ 74.0%

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

      3. distribute-rgt-neg-in 74.0%

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

   8. Simplified 74.0%

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

   if 5.0000000000000003e-34 < (*.f64 x y) < 0.050000000000000003

   1. Initial program 99.4%

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

   2. Taylor expanded in x around inf 76.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;y \cdot x \leq 0.05:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;y \cdot x \leq 10^{+53}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 3: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot z\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

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 (- (* y x) (* t z))))
   (if (<= t_1 (- INFINITY)) (- (/ 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 = (y * x) - (t * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		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 = (y * x) - (t * z);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		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 = (y * x) - (t * z)
	tmp = 0
	if t_1 <= -math.inf:
		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(y * x) - Float64(t * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		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 = (y * x) - (t * z);
	tmp = 0.0;
	if (t_1 <= -Inf)
		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[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], 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

   1. Initial program 67.6%

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

      1. div-sub 64.4%

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

      2. associate-/l* 76.2%

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

      3. associate-/l* 96.7%

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

   3. Applied egg-rr 96.7%

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

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

   1. Initial program 95.8%

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

4. Final simplification 95.9%

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

Alternative 4: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \frac{-y}{-a} - \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1e-28)
   (- (* x (/ (- y) (- a))) (/ t (/ a z)))
   (/ (- (* y x) (* t z)) a)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1e-28) {
		tmp = (x * (-y / -a)) - (t / (a / z));
	} else {
		tmp = ((y * x) - (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 (a <= (-1d-28)) then
        tmp = (x * (-y / -a)) - (t / (a / z))
    else
        tmp = ((y * x) - (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 (a <= -1e-28) {
		tmp = (x * (-y / -a)) - (t / (a / z));
	} else {
		tmp = ((y * x) - (t * z)) / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if a <= -1e-28:
		tmp = (x * (-y / -a)) - (t / (a / z))
	else:
		tmp = ((y * x) - (t * z)) / a
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1e-28)
		tmp = Float64(Float64(x * Float64(Float64(-y) / Float64(-a))) - Float64(t / Float64(a / z)));
	else
		tmp = Float64(Float64(Float64(y * x) - Float64(t * 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 (a <= -1e-28)
		tmp = (x * (-y / -a)) - (t / (a / z));
	else
		tmp = ((y * x) - (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[a, -1e-28], N[(N[(x * N[((-y) / (-a)), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.99999999999999971e-29

   1. Initial program 83.3%

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

      1. div-sub 83.3%

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

      2. *-un-lft-identity 83.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. times-frac 0.0%

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

      5. fma-neg 0.0%

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

      6. associate-/l* 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. fma-neg 0.0%

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

      2. associate-*l/ 0.0%

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

      3. *-lft-identity 0.0%

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

      4. associate-/l* 0.0%

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

      5. associate-/r/ 0.0%

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

      6. associate-*r/ 0.0%

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

      7. associate-/l* 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-/l* 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{a \cdot {\left(\sqrt{-1}\right)}^{2}}} - \frac{t}{\frac{a}{z}} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-*r/ 0.0%

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

      3. neg-mul-1 0.0%

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

      4. distribute-rgt-neg-in 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 88.8%

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

      8. neg-mul-1 88.8%

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

      9. associate-/l* 94.3%

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

      10. associate-/r/ 93.9%

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

   8. Simplified 93.9%

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

   if -9.99999999999999971e-29 < a

   1. Initial program 96.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \frac{-y}{-a} - \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \end{array} \]

Alternative 5: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+47} \lor \neg \left(y \cdot x \leq 10^{+53}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* y x) -1e+47) (not (<= (* y x) 1e+53)))
   (* 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 (((y * x) <= -1e+47) || !((y * x) <= 1e+53)) {
		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 (((y * x) <= (-1d+47)) .or. (.not. ((y * x) <= 1d+53))) 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 (((y * x) <= -1e+47) || !((y * x) <= 1e+53)) {
		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 ((y * x) <= -1e+47) or not ((y * x) <= 1e+53):
		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 ((Float64(y * x) <= -1e+47) || !(Float64(y * x) <= 1e+53))
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(t * Float64(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 (((y * x) <= -1e+47) || ~(((y * x) <= 1e+53)))
		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[Or[LessEqual[N[(y * x), $MachinePrecision], -1e+47], N[Not[LessEqual[N[(y * x), $MachinePrecision], 1e+53]], $MachinePrecision]], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e47 or 9.9999999999999999e52 < (*.f64 x y)

   1. Initial program 88.7%

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

   2. Taylor expanded in x around inf 78.1%

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

      1. associate-*r/ 83.0%

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

   4. Simplified 83.0%

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

   if -1e47 < (*.f64 x y) < 9.9999999999999999e52

   1. Initial program 95.0%

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

   2. Taylor expanded in x around 0 72.0%

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

      1. mul-1-neg 72.0%

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

      2. *-commutative 72.0%

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

      3. associate-*l/ 71.9%

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

      4. *-commutative 71.9%

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

      5. distribute-rgt-neg-in 71.9%

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

      6. distribute-frac-neg 71.9%

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

   4. Simplified 71.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+47} \lor \neg \left(y \cdot x \leq 10^{+53}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 6: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq 2 \cdot 10^{+246}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

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 x) 2e+246) (/ (- (* y x) (* t z)) a) (* x (/ y a))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * x) <= 2e+246) {
		tmp = ((y * x) - (t * z)) / 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 * x) <= 2d+246) then
        tmp = ((y * x) - (t * z)) / 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 * x) <= 2e+246) {
		tmp = ((y * x) - (t * z)) / 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 * x) <= 2e+246:
		tmp = ((y * x) - (t * z)) / 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(y * x) <= 2e+246)
		tmp = Float64(Float64(Float64(y * x) - Float64(t * z)) / 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 * x) <= 2e+246)
		tmp = ((y * x) - (t * z)) / 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[(y * x), $MachinePrecision], 2e+246], N[(N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.00000000000000014e246

   1. Initial program 93.6%

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

   if 2.00000000000000014e246 < (*.f64 x y)

   1. Initial program 77.4%

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

   2. Taylor expanded in x around inf 77.6%

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

      1. associate-*r/ 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 2 \cdot 10^{+246}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-134}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6.6e-134) (/ (* 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 (t <= 6.6e-134) {
		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 (t <= 6.6d-134) 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 (t <= 6.6e-134) {
		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 t <= 6.6e-134:
		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 (t <= 6.6e-134)
		tmp = Float64(Float64(y * 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 (t <= 6.6e-134)
		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[t, 6.6e-134], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.60000000000000038e-134

   1. Initial program 92.8%

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

   2. Taylor expanded in x around inf 57.2%

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

   if 6.60000000000000038e-134 < t

   1. Initial program 91.3%

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

   2. Taylor expanded in x around inf 47.7%

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

      1. associate-*r/ 49.8%

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

   4. Simplified 49.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-134}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8: 52.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

2. Taylor expanded in x around inf 53.9%

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

   1. associate-*r/ 53.2%

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

4. Simplified 53.2%

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

5. Final simplification 53.2%

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

Developer target: 91.2% 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 2023290 
(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))