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

Percentage Accurate: 91.1% → 96.1%

Time: 9.6s

Alternatives: 12

Speedup: 0.6×

• 27.0% 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: 91.1% 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: 96.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t}, -z\right)}{\sqrt{a\_m}} \cdot \frac{t}{\sqrt{a\_m}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_1 -1e+286)
      (* (/ (fma x (/ y t) (- z)) (sqrt a_m)) (/ t (sqrt a_m)))
      (if (<= t_1 4e+285) (/ t_1 a_m) (* z (/ (- (* x (/ y z)) t) a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -1e+286) {
		tmp = (fma(x, (y / t), -z) / sqrt(a_m)) * (t / sqrt(a_m));
	} else if (t_1 <= 4e+285) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e+286)
		tmp = Float64(Float64(fma(x, Float64(y / t), Float64(-z)) / sqrt(a_m)) * Float64(t / sqrt(a_m)));
	elseif (t_1 <= 4e+285)
		tmp = Float64(t_1 / a_m);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -1e+286], N[(N[(N[(x * N[(y / t), $MachinePrecision] + (-z)), $MachinePrecision] / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+285], N[(t$95$1 / a$95$m), $MachinePrecision], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.00000000000000003e286

   1. Initial program 75.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 75.4%

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

      1. associate-/l* 75.4%

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

   5. Simplified 75.4%

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

      1. *-commutative 75.4%

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

      2. add-sqr-sqrt 35.7%

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

      3. times-frac 48.0%

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

      4. fma-neg 48.0%

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

   7. Applied egg-rr 48.0%

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

   if -1.00000000000000003e286 < (-.f64 (*.f64 x y) (*.f64 z t)) < 3.9999999999999999e285

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 3.9999999999999999e285 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 64.6%

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

      1. div-sub 62.0%

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

      2. *-un-lft-identity 62.0%

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

      3. add-sqr-sqrt 29.8%

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

      4. times-frac 29.8%

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

      5. fma-neg 29.8%

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

      6. associate-/l* 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. fma-undefine 37.5%

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

      2. distribute-lft-neg-in 37.5%

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

      3. cancel-sign-sub-inv 37.5%

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

      4. associate-/l* 39.9%

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

      5. associate-*r/ 32.3%

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

      6. *-commutative 32.3%

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

      7. associate-/l* 37.2%

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

   6. Simplified 37.2%

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

   7. Taylor expanded in z around inf 82.0%

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

      1. times-frac 87.2%

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

      2. associate-*l/ 92.5%

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

      3. div-sub 95.2%

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

   9. Simplified 95.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{t}, -z\right)}{\sqrt{a}} \cdot \frac{t}{\sqrt{a}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{-t}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot t}{-a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* x y) -2e+39)
    (* x (/ y a_m))
    (if (<= (* x y) -1e-46)
      (* z (/ (- t) a_m))
      (if (or (<= (* x y) -2e-87) (not (<= (* x y) 5e-99)))
        (/ (* x y) a_m)
        (/ (* z t) (- a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * (y / a_m);
	} else if ((x * y) <= -1e-46) {
		tmp = z * (-t / a_m);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (z * t) / -a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((x * y) <= (-2d+39)) then
        tmp = x * (y / a_m)
    else if ((x * y) <= (-1d-46)) then
        tmp = z * (-t / a_m)
    else if (((x * y) <= (-2d-87)) .or. (.not. ((x * y) <= 5d-99))) then
        tmp = (x * y) / a_m
    else
        tmp = (z * t) / -a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -2e+39) {
		tmp = x * (y / a_m);
	} else if ((x * y) <= -1e-46) {
		tmp = z * (-t / a_m);
	} else if (((x * y) <= -2e-87) || !((x * y) <= 5e-99)) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (z * t) / -a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -2e+39:
		tmp = x * (y / a_m)
	elif (x * y) <= -1e-46:
		tmp = z * (-t / a_m)
	elif ((x * y) <= -2e-87) or not ((x * y) <= 5e-99):
		tmp = (x * y) / a_m
	else:
		tmp = (z * t) / -a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -2e+39)
		tmp = Float64(x * Float64(y / a_m));
	elseif (Float64(x * y) <= -1e-46)
		tmp = Float64(z * Float64(Float64(-t) / a_m));
	elseif ((Float64(x * y) <= -2e-87) || !(Float64(x * y) <= 5e-99))
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = Float64(Float64(z * t) / Float64(-a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -2e+39)
		tmp = x * (y / a_m);
	elseif ((x * y) <= -1e-46)
		tmp = z * (-t / a_m);
	elseif (((x * y) <= -2e-87) || ~(((x * y) <= 5e-99)))
		tmp = (x * y) / a_m;
	else
		tmp = (z * t) / -a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -2e+39], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-46], N[(z * N[((-t) / a$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-87], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-99]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(z * t), $MachinePrecision] / (-a$95$m)), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.99999999999999988e39

   1. Initial program 88.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-*r/ 73.4%

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

   5. Simplified 73.4%

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

   if -1.99999999999999988e39 < (*.f64 x y) < -1.00000000000000002e-46

   1. Initial program 94.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 74.6%

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

      1. *-commutative 74.6%

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

      2. associate-*r/ 74.2%

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

      3. neg-mul-1 74.2%

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

      4. distribute-rgt-neg-in 74.2%

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

      5. distribute-frac-neg 74.2%

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

   5. Simplified 74.2%

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

   if -1.00000000000000002e-46 < (*.f64 x y) < -2.00000000000000004e-87 or 4.99999999999999969e-99 < (*.f64 x y)

   1. Initial program 90.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 74.2%

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

   if -2.00000000000000004e-87 < (*.f64 x y) < 4.99999999999999969e-99

   1. Initial program 93.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.4%

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

      1. associate-*r* 83.4%

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

      2. mul-1-neg 83.4%

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

   5. Simplified 83.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-87} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{x}{a\_m} - z \cdot \frac{t}{y \cdot a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_1 (- INFINITY))
      (* y (- (/ x a_m) (* z (/ t (* y a_m)))))
      (if (<= t_1 5e+301) (/ t_1 a_m) (* z (/ (- (* x (/ y z)) t) a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * ((x / a_m) - (z * (t / (y * a_m))));
	} else if (t_1 <= 5e+301) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * ((x / a_m) - (z * (t / (y * a_m))));
	} else if (t_1 <= 5e+301) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * ((x / a_m) - (z * (t / (y * a_m))))
	elif t_1 <= 5e+301:
		tmp = t_1 / a_m
	else:
		tmp = z * (((x * (y / z)) - t) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(Float64(x / a_m) - Float64(z * Float64(t / Float64(y * a_m)))));
	elseif (t_1 <= 5e+301)
		tmp = Float64(t_1 / a_m);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * ((x / a_m) - (z * (t / (y * a_m))));
	elseif (t_1 <= 5e+301)
		tmp = t_1 / a_m;
	else
		tmp = z * (((x * (y / z)) - t) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(x / a$95$m), $MachinePrecision] - N[(z * N[(t / N[(y * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+301], N[(t$95$1 / a$95$m), $MachinePrecision], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-/l* 95.4%

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

   5. Simplified 95.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000004e301

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 5.0000000000000004e301 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 59.3%

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

      1. div-sub 56.3%

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

      2. *-un-lft-identity 56.3%

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

      3. add-sqr-sqrt 31.4%

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

      4. times-frac 31.4%

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

      5. fma-neg 31.4%

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

      6. associate-/l* 40.1%

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

   4. Applied egg-rr 40.1%

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

      1. fma-undefine 40.1%

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

      2. distribute-lft-neg-in 40.1%

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

      3. cancel-sign-sub-inv 40.1%

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

      4. associate-/l* 42.9%

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

      5. associate-*r/ 34.2%

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

      6. *-commutative 34.2%

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

      7. associate-/l* 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in z around inf 79.3%

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

      1. times-frac 85.3%

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

      2. associate-*l/ 91.4%

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

      3. div-sub 94.4%

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

   9. Simplified 94.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{x}{a} - z \cdot \frac{t}{y \cdot a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{a\_m} - t \cdot \frac{\frac{z}{y}}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (<= t_1 (- INFINITY))
      (* y (- (* x (/ 1.0 a_m)) (* t (/ (/ z y) a_m))))
      (if (<= t_1 5e+301) (/ t_1 a_m) (* z (/ (- (* x (/ y z)) t) a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * ((x * (1.0 / a_m)) - (t * ((z / y) / a_m)));
	} else if (t_1 <= 5e+301) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * ((x * (1.0 / a_m)) - (t * ((z / y) / a_m)));
	} else if (t_1 <= 5e+301) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * ((x * (1.0 / a_m)) - (t * ((z / y) / a_m)))
	elif t_1 <= 5e+301:
		tmp = t_1 / a_m
	else:
		tmp = z * (((x * (y / z)) - t) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(Float64(x * Float64(1.0 / a_m)) - Float64(t * Float64(Float64(z / y) / a_m))));
	elseif (t_1 <= 5e+301)
		tmp = Float64(t_1 / a_m);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * ((x * (1.0 / a_m)) - (t * ((z / y) / a_m)));
	elseif (t_1 <= 5e+301)
		tmp = t_1 / a_m;
	else
		tmp = z * (((x * (y / z)) - t) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(x * N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(t * N[(N[(z / y), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+301], N[(t$95$1 / a$95$m), $MachinePrecision], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-/l* 95.4%

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

   5. Simplified 95.4%

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

      1. div-inv 95.4%

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

      2. fma-neg 95.4%

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

      3. *-commutative 95.4%

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

   7. Applied egg-rr 95.4%

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

      1. fma-undefine 95.4%

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

      2. unsub-neg 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*l/ 81.6%

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

      5. associate-*r/ 95.4%

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

      6. associate-/r* 86.4%

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

   9. Simplified 86.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000004e301

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 5.0000000000000004e301 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 59.3%

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

      1. div-sub 56.3%

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

      2. *-un-lft-identity 56.3%

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

      3. add-sqr-sqrt 31.4%

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

      4. times-frac 31.4%

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

      5. fma-neg 31.4%

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

      6. associate-/l* 40.1%

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

   4. Applied egg-rr 40.1%

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

      1. fma-undefine 40.1%

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

      2. distribute-lft-neg-in 40.1%

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

      3. cancel-sign-sub-inv 40.1%

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

      4. associate-/l* 42.9%

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

      5. associate-*r/ 34.2%

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

      6. *-commutative 34.2%

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

      7. associate-/l* 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in z around inf 79.3%

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

      1. times-frac 85.3%

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

      2. associate-*l/ 91.4%

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

      3. div-sub 94.4%

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

   9. Simplified 94.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{a} - t \cdot \frac{\frac{z}{y}}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - t \cdot \frac{z}{x}}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* z t)) a_m)))
   (* a_s (if (<= t_1 2e+304) t_1 (* x (/ (- y (* t (/ z x))) a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - (z * t)) / a_m;
	double tmp;
	if (t_1 <= 2e+304) {
		tmp = t_1;
	} else {
		tmp = x * ((y - (t * (z / x))) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x * y) - (z * t)) / a_m
    if (t_1 <= 2d+304) then
        tmp = t_1
    else
        tmp = x * ((y - (t * (z / x))) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((x * y) - (z * t)) / a_m;
	double tmp;
	if (t_1 <= 2e+304) {
		tmp = t_1;
	} else {
		tmp = x * ((y - (t * (z / x))) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = ((x * y) - (z * t)) / a_m
	tmp = 0
	if t_1 <= 2e+304:
		tmp = t_1
	else:
		tmp = x * ((y - (t * (z / x))) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m)
	tmp = 0.0
	if (t_1 <= 2e+304)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(Float64(y - Float64(t * Float64(z / x))) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((x * y) - (z * t)) / a_m;
	tmp = 0.0;
	if (t_1 <= 2e+304)
		tmp = t_1;
	else
		tmp = x * ((y - (t * (z / x))) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 2e+304], t$95$1, N[(x * N[(N[(y - N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.9999999999999999e304

   1. Initial program 95.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1.9999999999999999e304 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 74.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.5%

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

   4. Taylor expanded in a around 0 81.4%

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

      1. associate-/l* 87.4%

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

      2. mul-1-neg 87.4%

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

      3. unsub-neg 87.4%

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

      4. associate-/l* 95.7%

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

   6. Simplified 95.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - t \cdot \frac{z}{x}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (* a_s (if (<= t_1 4e+285) (/ t_1 a_m) (* z (/ (- (* x (/ y z)) t) a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= 4e+285) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if (t_1 <= 4d+285) then
        tmp = t_1 / a_m
    else
        tmp = z * (((x * (y / z)) - t) / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= 4e+285) {
		tmp = t_1 / a_m;
	} else {
		tmp = z * (((x * (y / z)) - t) / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= 4e+285:
		tmp = t_1 / a_m
	else:
		tmp = z * (((x * (y / z)) - t) / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= 4e+285)
		tmp = Float64(t_1 / a_m);
	else
		tmp = Float64(z * Float64(Float64(Float64(x * Float64(y / z)) - t) / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= 4e+285)
		tmp = t_1 / a_m;
	else
		tmp = z * (((x * (y / z)) - t) / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 4e+285], N[(t$95$1 / a$95$m), $MachinePrecision], N[(z * N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < 3.9999999999999999e285

   1. Initial program 96.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 3.9999999999999999e285 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 64.6%

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

      1. div-sub 62.0%

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

      2. *-un-lft-identity 62.0%

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

      3. add-sqr-sqrt 29.8%

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

      4. times-frac 29.8%

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

      5. fma-neg 29.8%

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

      6. associate-/l* 37.5%

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

   4. Applied egg-rr 37.5%

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

      1. fma-undefine 37.5%

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

      2. distribute-lft-neg-in 37.5%

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

      3. cancel-sign-sub-inv 37.5%

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

      4. associate-/l* 39.9%

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

      5. associate-*r/ 32.3%

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

      6. *-commutative 32.3%

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

      7. associate-/l* 37.2%

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

   6. Simplified 37.2%

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

   7. Taylor expanded in z around inf 82.0%

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

      1. times-frac 87.2%

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

      2. associate-*l/ 92.5%

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

      3. div-sub 95.2%

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

   9. Simplified 95.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq 4 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x \cdot \frac{y}{z} - t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+37} \lor \neg \left(z \leq 9.8 \cdot 10^{-146}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= z -3.8e+37) (not (<= z 9.8e-146)))
    (* (- t) (/ z a_m))
    (/ (* x y) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z <= -3.8e+37) || !(z <= 9.8e-146)) {
		tmp = -t * (z / a_m);
	} else {
		tmp = (x * y) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((z <= (-3.8d+37)) .or. (.not. (z <= 9.8d-146))) then
        tmp = -t * (z / a_m)
    else
        tmp = (x * y) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z <= -3.8e+37) || !(z <= 9.8e-146)) {
		tmp = -t * (z / a_m);
	} else {
		tmp = (x * y) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (z <= -3.8e+37) or not (z <= 9.8e-146):
		tmp = -t * (z / a_m)
	else:
		tmp = (x * y) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((z <= -3.8e+37) || !(z <= 9.8e-146))
		tmp = Float64(Float64(-t) * Float64(z / a_m));
	else
		tmp = Float64(Float64(x * y) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((z <= -3.8e+37) || ~((z <= 9.8e-146)))
		tmp = -t * (z / a_m);
	else
		tmp = (x * y) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[z, -3.8e+37], N[Not[LessEqual[z, 9.8e-146]], $MachinePrecision]], N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999999e37 or 9.8000000000000008e-146 < z

   1. Initial program 87.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 60.9%

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

      1. mul-1-neg 60.9%

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

      2. associate-/l* 64.0%

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

      3. distribute-rgt-neg-in 64.0%

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

      4. distribute-neg-frac2 64.0%

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

   5. Simplified 64.0%

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

   if -3.7999999999999999e37 < z < 9.8000000000000008e-146

   1. Initial program 98.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 72.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+37} \lor \neg \left(z \leq 9.8 \cdot 10^{-146}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+37} \lor \neg \left(z \leq 1.42 \cdot 10^{-145}\right):\\ \;\;\;\;z \cdot \frac{-t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (or (<= z -2.15e+37) (not (<= z 1.42e-145)))
    (* z (/ (- t) a_m))
    (/ (* x y) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z <= -2.15e+37) || !(z <= 1.42e-145)) {
		tmp = z * (-t / a_m);
	} else {
		tmp = (x * y) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((z <= (-2.15d+37)) .or. (.not. (z <= 1.42d-145))) then
        tmp = z * (-t / a_m)
    else
        tmp = (x * y) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z <= -2.15e+37) || !(z <= 1.42e-145)) {
		tmp = z * (-t / a_m);
	} else {
		tmp = (x * y) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (z <= -2.15e+37) or not (z <= 1.42e-145):
		tmp = z * (-t / a_m)
	else:
		tmp = (x * y) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if ((z <= -2.15e+37) || !(z <= 1.42e-145))
		tmp = Float64(z * Float64(Float64(-t) / a_m));
	else
		tmp = Float64(Float64(x * y) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((z <= -2.15e+37) || ~((z <= 1.42e-145)))
		tmp = z * (-t / a_m);
	else
		tmp = (x * y) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[Or[LessEqual[z, -2.15e+37], N[Not[LessEqual[z, 1.42e-145]], $MachinePrecision]], N[(z * N[((-t) / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999998e37 or 1.4200000000000001e-145 < z

   1. Initial program 87.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-*r/ 62.4%

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

      3. neg-mul-1 62.4%

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

      4. distribute-rgt-neg-in 62.4%

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

      5. distribute-frac-neg 62.4%

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

   5. Simplified 62.4%

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

   if -2.1499999999999998e37 < z < 1.4200000000000001e-145

   1. Initial program 98.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 72.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+37} \lor \neg \left(z \leq 1.42 \cdot 10^{-145}\right):\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+200}:\\ \;\;\;\;z \cdot \frac{-t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* z t) -1e+200) (* z (/ (- t) a_m)) (/ (- (* x y) (* z t)) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= -1e+200) {
		tmp = z * (-t / a_m);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if ((z * t) <= (-1d+200)) then
        tmp = z * (-t / a_m)
    else
        tmp = ((x * y) - (z * t)) / a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= -1e+200) {
		tmp = z * (-t / a_m);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (z * t) <= -1e+200:
		tmp = z * (-t / a_m)
	else:
		tmp = ((x * y) - (z * t)) / a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(z * t) <= -1e+200)
		tmp = Float64(z * Float64(Float64(-t) / a_m));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((z * t) <= -1e+200)
		tmp = z * (-t / a_m);
	else
		tmp = ((x * y) - (z * t)) / a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -1e+200], N[(z * N[((-t) / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999997e199

   1. Initial program 74.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*r/ 100.0%

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

      3. neg-mul-1 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. distribute-frac-neg 100.0%

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

   5. Simplified 100.0%

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

   if -9.9999999999999997e199 < (*.f64 z t)

   1. Initial program 94.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+200}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (* a_s (if (<= y 1.5e-157) (* x (/ y a_m)) (* y (/ x a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (y <= 1.5e-157) {
		tmp = x * (y / a_m);
	} else {
		tmp = y * (x / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    real(8) :: tmp
    if (y <= 1.5d-157) then
        tmp = x * (y / a_m)
    else
        tmp = y * (x / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (y <= 1.5e-157) {
		tmp = x * (y / a_m);
	} else {
		tmp = y * (x / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if y <= 1.5e-157:
		tmp = x * (y / a_m)
	else:
		tmp = y * (x / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (y <= 1.5e-157)
		tmp = Float64(x * Float64(y / a_m));
	else
		tmp = Float64(y * Float64(x / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (y <= 1.5e-157)
		tmp = x * (y / a_m);
	else
		tmp = y * (x / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[y, 1.5e-157], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-157

   1. Initial program 92.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 52.9%

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

      1. associate-*r/ 52.3%

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

   5. Simplified 52.3%

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

   if 1.5e-157 < y

   1. Initial program 90.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 87.9%

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

      1. associate-/l* 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in t around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*r/ 49.9%

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

   8. Simplified 49.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \left(x \cdot \frac{y}{a\_m}\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (* x (/ y a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (x * (y / a_m));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    code = a_s * (x * (y / a_m))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (x * (y / a_m));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * (x * (y / a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(x * Float64(y / a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * (x * (y / a_m));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[\frac{x \cdot y - z \cdot t}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 51.7%

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

   1. associate-*r/ 50.3%

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

5. Simplified 50.3%

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

6. Final simplification 50.3%

   \[\leadsto x \cdot \frac{y}{a} \]
7. Add Preprocessing

Alternative 12: 49.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \frac{x \cdot y}{a\_m} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 1 a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (/ (* x y) a_m)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / a_m);
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    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_m
    code = a_s * ((x * y) / a_m)
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / a_m);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * ((x * y) / a_m)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(x * y) / a_m))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * ((x * y) / a_m);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[\frac{x \cdot y - z \cdot t}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 51.7%

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

4. Final simplification 51.7%

   \[\leadsto \frac{x \cdot y}{a} \]
5. Add Preprocessing

Developer target: 91.7% accurate, 0.4× 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 2024080 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (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))