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

Percentage Accurate: 91.6% → 95.0%

Time: 9.2s

Alternatives: 8

Speedup: 0.7×

• 27.5% 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: 91.6% 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: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.22 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, -z, x \cdot y\right) \cdot \frac{1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, \frac{z}{\frac{a\_m}{-t}}\right)\\ \end{array} \end{array} \]

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 1.22e+98)
    (* (fma t (- z) (* x y)) (/ 1.0 a_m))
    (fma (/ y a_m) x (/ z (/ a_m (- t)))))))

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 <= 1.22e+98) {
		tmp = fma(t, -z, (x * y)) * (1.0 / a_m);
	} else {
		tmp = fma((y / a_m), x, (z / (a_m / -t)));
	}
	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 <= 1.22e+98)
		tmp = Float64(fma(t, Float64(-z), Float64(x * y)) * Float64(1.0 / a_m));
	else
		tmp = fma(Float64(y / a_m), x, Float64(z / Float64(a_m / Float64(-t))));
	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, 1.22e+98], N[(N[(t * (-z) + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x + N[(z / N[(a$95$m / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.22e98

   1. Initial program 92.7%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 93.1

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

   4. Applied egg-rr 93.1%

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

   if 1.22e98 < a

   1. Initial program 81.3%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 85.6

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

   4. Applied egg-rr 85.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. distribute-frac-neg N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. /-lowering-/.f64 90.4

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

   6. Applied egg-rr 90.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, x, \color{blue}{\frac{-z}{\frac{a}{t}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.22 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(t, -z, x \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{z}{\frac{a}{-t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - t \cdot z\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \left(\frac{y}{a\_m} - t \cdot \frac{z}{a\_m \cdot x}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, \frac{z \cdot \left(-t\right)}{a\_m}\right)\\ \end{array} \end{array} \end{array} \]

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) (* t z))))
   (*
    a_s
    (if (<= t_1 (- INFINITY))
      (* x (- (/ y a_m) (* t (/ z (* a_m x)))))
      (if (<= t_1 2e+277)
        (/ t_1 a_m)
        (fma (/ x a_m) 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) - (t * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x * ((y / a_m) - (t * (z / (a_m * x))));
	} else if (t_1 <= 2e+277) {
		tmp = t_1 / a_m;
	} else {
		tmp = fma((x / a_m), 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(t * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x * Float64(Float64(y / a_m) - Float64(t * Float64(z / Float64(a_m * x)))));
	elseif (t_1 <= 2e+277)
		tmp = Float64(t_1 / a_m);
	else
		tmp = fma(Float64(x / a_m), y, Float64(Float64(z * Float64(-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[(t * z), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(x * N[(N[(y / a$95$m), $MachinePrecision] - N[(t * N[(z / N[(a$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+277], N[(t$95$1 / a$95$m), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y + N[(N[(z * (-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 60.5%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 60.5

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

   4. Applied egg-rr 60.5%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 90.7

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

   7. Simplified 90.7%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000001e277

   1. Initial program 98.8%

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

   if 2.00000000000000001e277 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 67.9%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 85.6

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

   4. Applied egg-rr 85.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - t \cdot z \leq -\infty:\\ \;\;\;\;x \cdot \left(\frac{y}{a} - t \cdot \frac{z}{a \cdot x}\right)\\ \mathbf{elif}\;x \cdot y - t \cdot z \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{z \cdot \left(-t\right)}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{a\_m}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, \frac{z \cdot \left(-t\right)}{a\_m}\right)\\ \end{array} \end{array} \]

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 (<= (* x y) (- INFINITY))
    (* y (* x (/ 1.0 a_m)))
    (if (<= (* x y) 2e+277)
      (/ (- (* x y) (* t z)) a_m)
      (fma (/ x a_m) 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) <= -((double) INFINITY)) {
		tmp = y * (x * (1.0 / a_m));
	} else if ((x * y) <= 2e+277) {
		tmp = ((x * y) - (t * z)) / a_m;
	} else {
		tmp = fma((x / a_m), 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) <= Float64(-Inf))
		tmp = Float64(y * Float64(x * Float64(1.0 / a_m)));
	elseif (Float64(x * y) <= 2e+277)
		tmp = Float64(Float64(Float64(x * y) - Float64(t * z)) / a_m);
	else
		tmp = fma(Float64(x / a_m), y, Float64(Float64(z * Float64(-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_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(y * N[(x * N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+277], N[(N[(N[(x * y), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y + N[(N[(z * (-t)), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   if -inf.0 < (*.f64 x y) < 2.00000000000000001e277

   1. Initial program 95.5%

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

   if 2.00000000000000001e277 < (*.f64 x y)

   1. Initial program 67.9%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{z \cdot \left(-t\right)}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-68}:\\ \;\;\;\;-z \cdot \frac{t}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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+48)
      t_1
      (if (<= (* x y) 1e-68)
        (- (* z (/ t a_m)))
        (if (<= (* x y) 4e+109) (/ (* 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+48) {
		tmp = t_1;
	} else if ((x * y) <= 1e-68) {
		tmp = -(z * (t / a_m));
	} else if ((x * y) <= 4e+109) {
		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+48)) then
        tmp = t_1
    else if ((x * y) <= 1d-68) then
        tmp = -(z * (t / a_m))
    else if ((x * y) <= 4d+109) 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+48) {
		tmp = t_1;
	} else if ((x * y) <= 1e-68) {
		tmp = -(z * (t / a_m));
	} else if ((x * y) <= 4e+109) {
		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+48:
		tmp = t_1
	elif (x * y) <= 1e-68:
		tmp = -(z * (t / a_m))
	elif (x * y) <= 4e+109:
		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+48)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-68)
		tmp = Float64(-Float64(z * Float64(t / a_m)));
	elseif (Float64(x * y) <= 4e+109)
		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+48)
		tmp = t_1;
	elseif ((x * y) <= 1e-68)
		tmp = -(z * (t / a_m));
	elseif ((x * y) <= 4e+109)
		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+48], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-68], (-N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), If[LessEqual[N[(x * y), $MachinePrecision], 4e+109], 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) < -4.00000000000000018e48 or 3.99999999999999993e109 < (*.f64 x y)

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 79.3

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

   7. Applied egg-rr 79.3%

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

   if -4.00000000000000018e48 < (*.f64 x y) < 1.00000000000000007e-68

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 75.3

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

   5. Simplified 75.3%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. neg-mul-1 N/A

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

      11. remove-double-neg N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-lowering-neg.f64 76.2

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

   7. Applied egg-rr 76.2%

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

   if 1.00000000000000007e-68 < (*.f64 x y) < 3.99999999999999993e109

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-68}:\\ \;\;\;\;-z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-68}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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+48)
      t_1
      (if (<= (* x y) 1e-68)
        (* t (/ z (- a_m)))
        (if (<= (* x y) 4e+109) (/ (* 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+48) {
		tmp = t_1;
	} else if ((x * y) <= 1e-68) {
		tmp = t * (z / -a_m);
	} else if ((x * y) <= 4e+109) {
		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+48)) then
        tmp = t_1
    else if ((x * y) <= 1d-68) then
        tmp = t * (z / -a_m)
    else if ((x * y) <= 4d+109) 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+48) {
		tmp = t_1;
	} else if ((x * y) <= 1e-68) {
		tmp = t * (z / -a_m);
	} else if ((x * y) <= 4e+109) {
		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+48:
		tmp = t_1
	elif (x * y) <= 1e-68:
		tmp = t * (z / -a_m)
	elif (x * y) <= 4e+109:
		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+48)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-68)
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	elseif (Float64(x * y) <= 4e+109)
		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+48)
		tmp = t_1;
	elseif ((x * y) <= 1e-68)
		tmp = t * (z / -a_m);
	elseif ((x * y) <= 4e+109)
		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+48], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-68], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+109], 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) < -4.00000000000000018e48 or 3.99999999999999993e109 < (*.f64 x y)

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 79.3

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

   7. Applied egg-rr 79.3%

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

   if -4.00000000000000018e48 < (*.f64 x y) < 1.00000000000000007e-68

   1. Initial program 93.8%

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

      1. div-inv N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 93.7

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 73.8

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

   7. Simplified 73.8%

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

   if 1.00000000000000007e-68 < (*.f64 x y) < 3.99999999999999993e109

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-68}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 0.7× speedup

Math

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

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 (<= (* x y) (- INFINITY))
    (* y (* x (/ 1.0 a_m)))
    (/ (- (* x y) (* 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 ((x * y) <= -((double) INFINITY)) {
		tmp = y * (x * (1.0 / a_m));
	} else {
		tmp = ((x * y) - (t * z)) / 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 tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x * (1.0 / a_m));
	} else {
		tmp = ((x * y) - (t * z)) / 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) <= -math.inf:
		tmp = y * (x * (1.0 / a_m))
	else:
		tmp = ((x * y) - (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 (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(y * Float64(x * Float64(1.0 / a_m)));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(t * z)) / 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) <= -Inf)
		tmp = y * (x * (1.0 / a_m));
	else
		tmp = ((x * y) - (t * z)) / 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], (-Infinity)], N[(y * N[(x * N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 99.6

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

   7. Applied egg-rr 99.6%

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

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

   1. Initial program 93.7%

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

4. Final simplification 94.2%

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

Alternative 7: 51.8% accurate, 1.5× 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 90.9%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 48.7

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

5. Simplified 48.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 50.2

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

7. Applied egg-rr 50.2%

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

8. Final simplification 50.2%

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

Alternative 8: 52.1% accurate, 1.5× speedup

Math

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

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 (* 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) {
	return a_s * (y * (x / 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 * (y * (x / 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 * (y * (x / 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 * (y * (x / 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(y * Float64(x / 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 * (y * (x / 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[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 48.7

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

5. Simplified 48.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 50.8

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

7. Applied egg-rr 50.8%

   \[\leadsto \color{blue}{y \cdot \frac{x}{a}} \]
8. Add Preprocessing

Developer Target 1: 92.2% accurate, 0.5× 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 2024199 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

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