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

Percentage Accurate: 91.0% → 95.6%

Time: 8.6s

Alternatives: 11

Speedup: 0.4×

• 28.2% 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: 91.0% 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: 95.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot {\left(\sqrt{\frac{y - t \cdot \frac{z}{x}}{a\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a\_m}, z \cdot \frac{t}{-a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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))
      (* x (pow (sqrt (/ (- y (* t (/ z x))) a_m)) 2.0))
      (if (<= t_1 5e+267)
        (/ (fma x y (* z (- t))) a_m)
        (fma y (/ x 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x * pow(sqrt(((y - (t * (z / x))) / a_m)), 2.0);
	} else if (t_1 <= 5e+267) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = fma(y, (x / a_m), (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(x * (sqrt(Float64(Float64(y - Float64(t * Float64(z / x))) / a_m)) ^ 2.0));
	elseif (t_1 <= 5e+267)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = fma(y, Float64(x / a_m), Float64(z * Float64(t / Float64(-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, (-Infinity)], N[(x * N[Power[N[Sqrt[N[(N[(y - N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+267], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision] + N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. associate-/l* 89.7%

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

      5. *-commutative 89.7%

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

      6. associate-/r* 93.3%

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

   5. Simplified 93.3%

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

      1. add-sqr-sqrt 37.5%

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

      2. pow2 37.5%

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

      3. associate-*r/ 37.5%

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

      4. sub-div 37.5%

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

   7. Applied egg-rr 37.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e267

   1. Initial program 98.8%

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

      1. div-sub 97.2%

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

      2. *-commutative 97.2%

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

      3. div-sub 98.8%

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

      4. *-commutative 98.8%

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

      5. fmm-def 98.8%

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

      6. distribute-rgt-neg-out 98.8%

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

   3. Simplified 98.8%

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

   if 4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.0%

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

      1. div-sub 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-/l* 72.5%

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

      4. fmm-def 75.0%

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

      5. associate-/l* 87.2%

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

   4. Applied egg-rr 87.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot {\left(\sqrt{\frac{y - t \cdot \frac{z}{x}}{a}}\right)}^{2}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, z \cdot \frac{t}{-a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \left(\frac{y}{a\_m} - t \cdot \frac{\frac{z}{x}}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a\_m}, z \cdot \frac{t}{-a\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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))
      (* x (- (/ y a_m) (* t (/ (/ z x) a_m))))
      (if (<= t_1 5e+267)
        (/ (fma x y (* z (- t))) a_m)
        (fma y (/ x 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x * ((y / a_m) - (t * ((z / x) / a_m)));
	} else if (t_1 <= 5e+267) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = fma(y, (x / a_m), (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(x * Float64(Float64(y / a_m) - Float64(t * Float64(Float64(z / x) / a_m))));
	elseif (t_1 <= 5e+267)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = fma(y, Float64(x / a_m), Float64(z * Float64(t / Float64(-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, (-Infinity)], N[(x * N[(N[(y / a$95$m), $MachinePrecision] - N[(t * N[(N[(z / x), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+267], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision] + N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. associate-/l* 89.7%

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

      5. *-commutative 89.7%

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

      6. associate-/r* 93.3%

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

   5. Simplified 93.3%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e267

   1. Initial program 98.8%

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

      1. div-sub 97.2%

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

      2. *-commutative 97.2%

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

      3. div-sub 98.8%

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

      4. *-commutative 98.8%

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

      5. fmm-def 98.8%

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

      6. distribute-rgt-neg-out 98.8%

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

   3. Simplified 98.8%

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

   if 4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.0%

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

      1. div-sub 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-/l* 72.5%

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

      4. fmm-def 75.0%

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

      5. associate-/l* 87.2%

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

   4. Applied egg-rr 87.2%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \left(\frac{y}{a} - t \cdot \frac{\frac{z}{x}}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, z \cdot \frac{t}{-a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a\_m}{x}} - t \cdot \frac{z}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= a_m 7e-30)
    (/ (fma x y (* z (- t))) a_m)
    (- (/ y (/ a_m x)) (* t (/ z 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 (a_m <= 7e-30) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = (y / (a_m / x)) - (t * (z / 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 (a_m <= 7e-30)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = Float64(Float64(y / Float64(a_m / x)) - Float64(t * Float64(z / 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_] := N[(a$95$s * If[LessEqual[a$95$m, 7e-30], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / N[(a$95$m / x), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 7.0000000000000006e-30

   1. Initial program 91.4%

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

      1. div-sub 88.7%

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

      2. *-commutative 88.7%

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

      3. div-sub 91.4%

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

      4. *-commutative 91.4%

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

      5. fmm-def 92.5%

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

      6. distribute-rgt-neg-out 92.5%

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

   3. Simplified 92.5%

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

   if 7.0000000000000006e-30 < a

   1. Initial program 88.3%

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

      1. div-sub 88.3%

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

      2. *-un-lft-identity 88.3%

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

      3. add-sqr-sqrt 88.1%

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

      4. times-frac 88.1%

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

      5. fmm-def 88.1%

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

      6. associate-/l* 89.3%

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

   4. Applied egg-rr 89.3%

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

      1. fmm-undef 89.3%

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

      2. associate-*l/ 89.3%

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

      3. *-lft-identity 89.3%

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

      4. associate-/l* 93.0%

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

      5. associate-*l/ 92.9%

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

      6. associate-*r/ 91.7%

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

      7. *-commutative 91.7%

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

      8. associate-/l* 92.8%

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

   6. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. frac-times 91.9%

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

      3. *-un-lft-identity 91.9%

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

   8. Applied egg-rr 91.9%

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

      1. associate-*l/ 91.9%

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

      2. rem-square-sqrt 92.0%

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

   10. Simplified 92.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}} - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-36}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 a_m))))
   (*
    a_s
    (if (<= (* x y) -4e-14)
      t_1
      (if (<= (* x y) 4e-155)
        (* t (/ z (- a_m)))
        (if (<= (* x y) 2e-71)
          t_1
          (if (<= (* x y) 2e-36)
            (* z (/ t (- a_m)))
            (if (<= (* x y) 4e+111) (/ (* x y) a_m) t_1))))))))

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 / a_m);
	double tmp;
	if ((x * y) <= -4e-14) {
		tmp = t_1;
	} else if ((x * y) <= 4e-155) {
		tmp = t * (z / -a_m);
	} else if ((x * y) <= 2e-71) {
		tmp = t_1;
	} else if ((x * y) <= 2e-36) {
		tmp = z * (t / -a_m);
	} else if ((x * y) <= 4e+111) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	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 / a_m)
    if ((x * y) <= (-4d-14)) then
        tmp = t_1
    else if ((x * y) <= 4d-155) then
        tmp = t * (z / -a_m)
    else if ((x * y) <= 2d-71) then
        tmp = t_1
    else if ((x * y) <= 2d-36) then
        tmp = z * (t / -a_m)
    else if ((x * y) <= 4d+111) then
        tmp = (x * y) / a_m
    else
        tmp = t_1
    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 / a_m);
	double tmp;
	if ((x * y) <= -4e-14) {
		tmp = t_1;
	} else if ((x * y) <= 4e-155) {
		tmp = t * (z / -a_m);
	} else if ((x * y) <= 2e-71) {
		tmp = t_1;
	} else if ((x * y) <= 2e-36) {
		tmp = z * (t / -a_m);
	} else if ((x * y) <= 4e+111) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	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 / a_m)
	tmp = 0
	if (x * y) <= -4e-14:
		tmp = t_1
	elif (x * y) <= 4e-155:
		tmp = t * (z / -a_m)
	elif (x * y) <= 2e-71:
		tmp = t_1
	elif (x * y) <= 2e-36:
		tmp = z * (t / -a_m)
	elif (x * y) <= 4e+111:
		tmp = (x * y) / a_m
	else:
		tmp = t_1
	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(x * Float64(y / a_m))
	tmp = 0.0
	if (Float64(x * y) <= -4e-14)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-155)
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	elseif (Float64(x * y) <= 2e-71)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-36)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	elseif (Float64(x * y) <= 4e+111)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = t_1;
	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 / a_m);
	tmp = 0.0;
	if ((x * y) <= -4e-14)
		tmp = t_1;
	elseif ((x * y) <= 4e-155)
		tmp = t * (z / -a_m);
	elseif ((x * y) <= 2e-71)
		tmp = t_1;
	elseif ((x * y) <= 2e-36)
		tmp = z * (t / -a_m);
	elseif ((x * y) <= 4e+111)
		tmp = (x * y) / a_m;
	else
		tmp = t_1;
	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[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -4e-14], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-155], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-71], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-36], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+111], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4e-14 or 4.00000000000000006e-155 < (*.f64 x y) < 1.9999999999999998e-71 or 3.99999999999999983e111 < (*.f64 x y)

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf 76.9%

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

      1. associate-*r/ 81.0%

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

   5. Simplified 81.0%

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

   if -4e-14 < (*.f64 x y) < 4.00000000000000006e-155

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

      2. associate-/l* 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. distribute-neg-frac2 74.4%

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

   5. Simplified 74.4%

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

   if 1.9999999999999998e-71 < (*.f64 x y) < 1.9999999999999999e-36

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*r/ 65.9%

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

      4. distribute-rgt-neg-in 65.9%

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

      5. distribute-frac-neg 65.9%

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

   5. Simplified 65.9%

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

   if 1.9999999999999999e-36 < (*.f64 x y) < 3.99999999999999983e111

   1. Initial program 96.2%

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

   3. Taylor expanded in x around inf 66.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-36}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{z \cdot t}{-a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-36}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 a_m))))
   (*
    a_s
    (if (<= (* x y) -4e-14)
      t_1
      (if (<= (* x y) 4e-155)
        (/ (* z t) (- a_m))
        (if (<= (* x y) 2e-71)
          t_1
          (if (<= (* x y) 2e-36)
            (* z (/ t (- a_m)))
            (if (<= (* x y) 4e+111) (/ (* x y) a_m) t_1))))))))

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 / a_m);
	double tmp;
	if ((x * y) <= -4e-14) {
		tmp = t_1;
	} else if ((x * y) <= 4e-155) {
		tmp = (z * t) / -a_m;
	} else if ((x * y) <= 2e-71) {
		tmp = t_1;
	} else if ((x * y) <= 2e-36) {
		tmp = z * (t / -a_m);
	} else if ((x * y) <= 4e+111) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	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 / a_m)
    if ((x * y) <= (-4d-14)) then
        tmp = t_1
    else if ((x * y) <= 4d-155) then
        tmp = (z * t) / -a_m
    else if ((x * y) <= 2d-71) then
        tmp = t_1
    else if ((x * y) <= 2d-36) then
        tmp = z * (t / -a_m)
    else if ((x * y) <= 4d+111) then
        tmp = (x * y) / a_m
    else
        tmp = t_1
    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 / a_m);
	double tmp;
	if ((x * y) <= -4e-14) {
		tmp = t_1;
	} else if ((x * y) <= 4e-155) {
		tmp = (z * t) / -a_m;
	} else if ((x * y) <= 2e-71) {
		tmp = t_1;
	} else if ((x * y) <= 2e-36) {
		tmp = z * (t / -a_m);
	} else if ((x * y) <= 4e+111) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	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 / a_m)
	tmp = 0
	if (x * y) <= -4e-14:
		tmp = t_1
	elif (x * y) <= 4e-155:
		tmp = (z * t) / -a_m
	elif (x * y) <= 2e-71:
		tmp = t_1
	elif (x * y) <= 2e-36:
		tmp = z * (t / -a_m)
	elif (x * y) <= 4e+111:
		tmp = (x * y) / a_m
	else:
		tmp = t_1
	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(x * Float64(y / a_m))
	tmp = 0.0
	if (Float64(x * y) <= -4e-14)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-155)
		tmp = Float64(Float64(z * t) / Float64(-a_m));
	elseif (Float64(x * y) <= 2e-71)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-36)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	elseif (Float64(x * y) <= 4e+111)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = t_1;
	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 / a_m);
	tmp = 0.0;
	if ((x * y) <= -4e-14)
		tmp = t_1;
	elseif ((x * y) <= 4e-155)
		tmp = (z * t) / -a_m;
	elseif ((x * y) <= 2e-71)
		tmp = t_1;
	elseif ((x * y) <= 2e-36)
		tmp = z * (t / -a_m);
	elseif ((x * y) <= 4e+111)
		tmp = (x * y) / a_m;
	else
		tmp = t_1;
	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[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -4e-14], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-155], N[(N[(z * t), $MachinePrecision] / (-a$95$m)), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-71], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-36], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+111], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4e-14 or 4.00000000000000006e-155 < (*.f64 x y) < 1.9999999999999998e-71 or 3.99999999999999983e111 < (*.f64 x y)

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf 76.9%

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

      1. associate-*r/ 81.0%

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

   5. Simplified 81.0%

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

   if -4e-14 < (*.f64 x y) < 4.00000000000000006e-155

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0 81.9%

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

      1. associate-*r* 81.9%

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

      2. mul-1-neg 81.9%

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

   5. Simplified 81.9%

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

   if 1.9999999999999998e-71 < (*.f64 x y) < 1.9999999999999999e-36

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*r/ 65.9%

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

      4. distribute-rgt-neg-in 65.9%

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

      5. distribute-frac-neg 65.9%

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

   5. Simplified 65.9%

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

   if 1.9999999999999999e-36 < (*.f64 x y) < 3.99999999999999983e111

   1. Initial program 96.2%

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

   3. Taylor expanded in x around inf 66.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-155}:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-36}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+234} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+267}\right):\\ \;\;\;\;x \cdot \frac{y}{a\_m} - z \cdot \frac{t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 (or (<= t_1 -5e+234) (not (<= t_1 5e+267)))
      (- (* x (/ y a_m)) (* z (/ t a_m)))
      (/ t_1 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 <= -5e+234) || !(t_1 <= 5e+267)) {
		tmp = (x * (y / a_m)) - (z * (t / a_m));
	} else {
		tmp = t_1 / 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 <= (-5d+234)) .or. (.not. (t_1 <= 5d+267))) then
        tmp = (x * (y / a_m)) - (z * (t / a_m))
    else
        tmp = t_1 / 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 <= -5e+234) || !(t_1 <= 5e+267)) {
		tmp = (x * (y / a_m)) - (z * (t / a_m));
	} else {
		tmp = t_1 / 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 <= -5e+234) or not (t_1 <= 5e+267):
		tmp = (x * (y / a_m)) - (z * (t / a_m))
	else:
		tmp = t_1 / 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 <= -5e+234) || !(t_1 <= 5e+267))
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(z * Float64(t / a_m)));
	else
		tmp = Float64(t_1 / 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 <= -5e+234) || ~((t_1 <= 5e+267)))
		tmp = (x * (y / a_m)) - (z * (t / a_m));
	else
		tmp = t_1 / 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[Or[LessEqual[t$95$1, -5e+234], N[Not[LessEqual[t$95$1, 5e+267]], $MachinePrecision]], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000003e234 or 4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 72.2%

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

      1. div-sub 69.7%

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

      2. associate-/l* 82.9%

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

      3. associate-/l* 92.2%

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

   4. Applied egg-rr 92.2%

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

   if -5.0000000000000003e234 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e267

   1. Initial program 98.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+234} \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+267}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+234}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}} - t \cdot \frac{z}{a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m} - z \cdot \frac{t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 -5e+234)
      (- (/ x (/ a_m y)) (* t (/ z a_m)))
      (if (<= t_1 5e+267) (/ t_1 a_m) (- (* 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -5e+234) {
		tmp = (x / (a_m / y)) - (t * (z / a_m));
	} else if (t_1 <= 5e+267) {
		tmp = t_1 / a_m;
	} else {
		tmp = (x * (y / a_m)) - (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 <= (-5d+234)) then
        tmp = (x / (a_m / y)) - (t * (z / a_m))
    else if (t_1 <= 5d+267) then
        tmp = t_1 / a_m
    else
        tmp = (x * (y / a_m)) - (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 <= -5e+234) {
		tmp = (x / (a_m / y)) - (t * (z / a_m));
	} else if (t_1 <= 5e+267) {
		tmp = t_1 / a_m;
	} else {
		tmp = (x * (y / a_m)) - (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 <= -5e+234:
		tmp = (x / (a_m / y)) - (t * (z / a_m))
	elif t_1 <= 5e+267:
		tmp = t_1 / a_m
	else:
		tmp = (x * (y / a_m)) - (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 <= -5e+234)
		tmp = Float64(Float64(x / Float64(a_m / y)) - Float64(t * Float64(z / a_m)));
	elseif (t_1 <= 5e+267)
		tmp = Float64(t_1 / a_m);
	else
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(z * Float64(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 <= -5e+234)
		tmp = (x / (a_m / y)) - (t * (z / a_m));
	elseif (t_1 <= 5e+267)
		tmp = t_1 / a_m;
	else
		tmp = (x * (y / a_m)) - (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, -5e+234], N[(N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+267], N[(t$95$1 / a$95$m), $MachinePrecision], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000003e234

   1. Initial program 81.2%

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

      1. div-sub 78.7%

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

      2. *-un-lft-identity 78.7%

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

      3. add-sqr-sqrt 33.4%

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

      4. times-frac 33.5%

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

      5. fmm-def 33.5%

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

      6. associate-/l* 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. fmm-undef 37.8%

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

      2. associate-*l/ 37.8%

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

      3. *-lft-identity 37.8%

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

      4. associate-/l* 42.5%

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

      5. associate-*l/ 44.8%

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

      6. associate-*r/ 40.4%

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

      7. *-commutative 40.4%

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

      8. associate-/l* 44.8%

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

   6. Simplified 44.8%

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

      1. *-commutative 44.8%

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

      2. clear-num 44.8%

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

      3. frac-times 44.9%

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

      4. *-un-lft-identity 44.9%

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

   8. Applied egg-rr 44.9%

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

      1. associate-*l/ 44.9%

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

      2. rem-square-sqrt 97.3%

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

   10. Simplified 97.3%

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

   if -5.0000000000000003e234 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e267

   1. Initial program 98.7%

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

   if 4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.0%

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

      1. div-sub 60.5%

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

      2. associate-/l* 75.0%

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

      3. associate-/l* 87.1%

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

   4. Applied egg-rr 87.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+234}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \left(\frac{y}{a\_m} - t \cdot \frac{\frac{z}{x}}{a\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m} - z \cdot \frac{t}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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))
      (* x (- (/ y a_m) (* t (/ (/ z x) a_m))))
      (if (<= t_1 5e+267) (/ t_1 a_m) (- (* 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 t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x * ((y / a_m) - (t * ((z / x) / a_m)));
	} else if (t_1 <= 5e+267) {
		tmp = t_1 / a_m;
	} else {
		tmp = (x * (y / a_m)) - (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 = x * ((y / a_m) - (t * ((z / x) / a_m)));
	} else if (t_1 <= 5e+267) {
		tmp = t_1 / a_m;
	} else {
		tmp = (x * (y / a_m)) - (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 = x * ((y / a_m) - (t * ((z / x) / a_m)))
	elif t_1 <= 5e+267:
		tmp = t_1 / a_m
	else:
		tmp = (x * (y / a_m)) - (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(x * Float64(Float64(y / a_m) - Float64(t * Float64(Float64(z / x) / a_m))));
	elseif (t_1 <= 5e+267)
		tmp = Float64(t_1 / a_m);
	else
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(z * Float64(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 = x * ((y / a_m) - (t * ((z / x) / a_m)));
	elseif (t_1 <= 5e+267)
		tmp = t_1 / a_m;
	else
		tmp = (x * (y / a_m)) - (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[(x * N[(N[(y / a$95$m), $MachinePrecision] - N[(t * N[(N[(z / x), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+267], N[(t$95$1 / a$95$m), $MachinePrecision], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. associate-/l* 89.7%

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

      5. *-commutative 89.7%

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

      6. associate-/r* 93.3%

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

   5. Simplified 93.3%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e267

   1. Initial program 98.8%

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

   if 4.9999999999999999e267 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 63.0%

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

      1. div-sub 60.5%

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

      2. associate-/l* 75.0%

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

      3. associate-/l* 87.1%

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

   4. Applied egg-rr 87.1%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \left(\frac{y}{a} - t \cdot \frac{\frac{z}{x}}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 a_m))))
   (*
    a_s
    (if (<= (* x y) -4e-14)
      t_1
      (if (<= (* x y) 4e-155)
        (* t (/ z (- a_m)))
        (if (<= (* x y) 4e+111) (/ (* x y) a_m) t_1))))))

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 / a_m);
	double tmp;
	if ((x * y) <= -4e-14) {
		tmp = t_1;
	} else if ((x * y) <= 4e-155) {
		tmp = t * (z / -a_m);
	} else if ((x * y) <= 4e+111) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	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 / a_m)
    if ((x * y) <= (-4d-14)) then
        tmp = t_1
    else if ((x * y) <= 4d-155) then
        tmp = t * (z / -a_m)
    else if ((x * y) <= 4d+111) then
        tmp = (x * y) / a_m
    else
        tmp = t_1
    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 / a_m);
	double tmp;
	if ((x * y) <= -4e-14) {
		tmp = t_1;
	} else if ((x * y) <= 4e-155) {
		tmp = t * (z / -a_m);
	} else if ((x * y) <= 4e+111) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1;
	}
	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 / a_m)
	tmp = 0
	if (x * y) <= -4e-14:
		tmp = t_1
	elif (x * y) <= 4e-155:
		tmp = t * (z / -a_m)
	elif (x * y) <= 4e+111:
		tmp = (x * y) / a_m
	else:
		tmp = t_1
	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(x * Float64(y / a_m))
	tmp = 0.0
	if (Float64(x * y) <= -4e-14)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e-155)
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	elseif (Float64(x * y) <= 4e+111)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = t_1;
	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 / a_m);
	tmp = 0.0;
	if ((x * y) <= -4e-14)
		tmp = t_1;
	elseif ((x * y) <= 4e-155)
		tmp = t * (z / -a_m);
	elseif ((x * y) <= 4e+111)
		tmp = (x * y) / a_m;
	else
		tmp = t_1;
	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[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -4e-14], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e-155], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+111], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4e-14 or 3.99999999999999983e111 < (*.f64 x y)

   1. Initial program 84.0%

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

   3. Taylor expanded in x around inf 76.1%

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

      1. associate-*r/ 80.5%

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

   5. Simplified 80.5%

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

   if -4e-14 < (*.f64 x y) < 4.00000000000000006e-155

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

      2. associate-/l* 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. distribute-neg-frac2 74.4%

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

   5. Simplified 74.4%

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

   if 4.00000000000000006e-155 < (*.f64 x y) < 3.99999999999999983e111

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf 61.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+234} \lor \neg \left(x \cdot y \leq 10^{+164}\right):\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= (* x y) -5e+234) (not (<= (* x y) 1e+164)))
    (* x (/ y 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 (((x * y) <= -5e+234) || !((x * y) <= 1e+164)) {
		tmp = x * (y / 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 (((x * y) <= (-5d+234)) .or. (.not. ((x * y) <= 1d+164))) then
        tmp = x * (y / 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 (((x * y) <= -5e+234) || !((x * y) <= 1e+164)) {
		tmp = x * (y / 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 ((x * y) <= -5e+234) or not ((x * y) <= 1e+164):
		tmp = x * (y / 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(x * y) <= -5e+234) || !(Float64(x * y) <= 1e+164))
		tmp = Float64(x * Float64(y / 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 (((x * y) <= -5e+234) || ~(((x * y) <= 1e+164)))
		tmp = x * (y / 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[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+234], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+164]], $MachinePrecision]], N[(x * N[(y / 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 x y) < -5.0000000000000003e234 or 1e164 < (*.f64 x y)

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 76.0%

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

      1. associate-*r/ 90.2%

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

   5. Simplified 90.2%

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

   if -5.0000000000000003e234 < (*.f64 x y) < 1e164

   1. Initial program 95.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+234} \lor \neg \left(x \cdot y \leq 10^{+164}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.5% accurate, 1.8× speedup

Math

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

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) 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 90.5%

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

3. Taylor expanded in x around inf 52.8%

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

   1. associate-*r/ 54.0%

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

5. Simplified 54.0%

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

6. Final simplification 54.0%

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

Developer target: 91.5% 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 2024089 
(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))