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

Percentage Accurate: 91.7% → 94.6%

Time: 10.2s

Alternatives: 11

Speedup: 0.6×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - (z * t)) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% 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: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, z \cdot \frac{t}{-a\_m}\right)\\ \end{array} \end{array} \]

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 (<= a_m 8.4e+68)
    (/ (- (* x y) (* z t)) a_m)
    (fma (/ y a_m) x (* 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 (a_m <= 8.4e+68) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma((y / a_m), x, (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 (a_m <= 8.4e+68)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(y / a_m), x, 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_] := N[(a$95$s * If[LessEqual[a$95$m, 8.4e+68], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x + N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.40000000000000003e68

   1. Initial program 91.7%

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

   if 8.40000000000000003e68 < a

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 90.4

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

   4. Applied egg-rr 90.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-neg.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 97.4

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

   6. Applied egg-rr 97.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, z \cdot \frac{t}{-a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;-\frac{z}{\frac{a\_m}{t}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \end{array} \end{array} \]

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 (<= (* z t) (- INFINITY))
    (- (/ z (/ a_m t)))
    (if (<= (* z t) 2e+188) (/ (- (* x y) (* z t)) a_m) (* 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 ((z * t) <= -((double) INFINITY)) {
		tmp = -(z / (a_m / t));
	} else if ((z * t) <= 2e+188) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = 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 ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = -(z / (a_m / t));
	} else if ((z * t) <= 2e+188) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = 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 (z * t) <= -math.inf:
		tmp = -(z / (a_m / t))
	elif (z * t) <= 2e+188:
		tmp = ((x * y) - (z * t)) / a_m
	else:
		tmp = 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(z * t) <= Float64(-Inf))
		tmp = Float64(-Float64(z / Float64(a_m / t)));
	elseif (Float64(z * t) <= 2e+188)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = -(z / (a_m / t));
	elseif ((z * t) <= 2e+188)
		tmp = ((x * y) - (z * t)) / a_m;
	else
		tmp = 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[(z * t), $MachinePrecision], (-Infinity)], (-N[(z / N[(a$95$m / t), $MachinePrecision]), $MachinePrecision]), If[LessEqual[N[(z * t), $MachinePrecision], 2e+188], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 57.0

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

   5. Simplified 57.0%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -inf.0 < (*.f64 z t) < 2e188

   1. Initial program 95.2%

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

   if 2e188 < (*.f64 z t)

   1. Initial program 75.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 75.0

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

   5. Simplified 75.0%

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

      1. lift-neg.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 92.6

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

   7. Applied egg-rr 92.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;-\frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;-\frac{z}{\frac{a\_m}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \end{array} \end{array} \]

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) -50000000000000.0)
    (* y (/ x a_m))
    (if (<= (* x y) 2e-23) (- (/ z (/ a_m t))) (/ x (/ a_m y))))))

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) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = -(z / (a_m / t));
	} else {
		tmp = x / (a_m / y);
	}
	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) <= (-50000000000000.0d0)) then
        tmp = y * (x / a_m)
    else if ((x * y) <= 2d-23) then
        tmp = -(z / (a_m / t))
    else
        tmp = x / (a_m / y)
    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) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = -(z / (a_m / t));
	} else {
		tmp = x / (a_m / y);
	}
	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) <= -50000000000000.0:
		tmp = y * (x / a_m)
	elif (x * y) <= 2e-23:
		tmp = -(z / (a_m / t))
	else:
		tmp = x / (a_m / y)
	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) <= -50000000000000.0)
		tmp = Float64(y * Float64(x / a_m));
	elseif (Float64(x * y) <= 2e-23)
		tmp = Float64(-Float64(z / Float64(a_m / t)));
	else
		tmp = Float64(x / Float64(a_m / y));
	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) <= -50000000000000.0)
		tmp = y * (x / a_m);
	elseif ((x * y) <= 2e-23)
		tmp = -(z / (a_m / t));
	else
		tmp = x / (a_m / y);
	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], -50000000000000.0], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-23], (-N[(z / N[(a$95$m / t), $MachinePrecision]), $MachinePrecision]), N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e13

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

   if -5e13 < (*.f64 x y) < 1.99999999999999992e-23

   1. Initial program 90.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 74.2

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

   5. Simplified 74.2%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if 1.99999999999999992e-23 < (*.f64 x y)

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.6

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

   5. Simplified 79.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.3

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

   7. Applied egg-rr 74.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;-\frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.5× speedup

Math

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

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))
    (* x (* y (/ 1.0 a_m)))
    (fma (- t) (/ z a_m) (/ (* x y) a_m)))))

C

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

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(x * Float64(y * Float64(1.0 / a_m)));
	else
		tmp = fma(Float64(-t), Float64(z / a_m), Float64(Float64(x * y) / 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[(x * N[(y * N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t) * N[(z / a$95$m), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 94.4

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

   4. Applied egg-rr 94.4%

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

4. Final simplification 94.8%

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

Alternative 5: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \end{array} \end{array} \]

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) -50000000000000.0)
    (* y (/ x a_m))
    (if (<= (* x y) 2e-23) (* z (/ t (- a_m))) (/ x (/ a_m y))))))

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) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = z * (t / -a_m);
	} else {
		tmp = x / (a_m / y);
	}
	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) <= (-50000000000000.0d0)) then
        tmp = y * (x / a_m)
    else if ((x * y) <= 2d-23) then
        tmp = z * (t / -a_m)
    else
        tmp = x / (a_m / y)
    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) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = z * (t / -a_m);
	} else {
		tmp = x / (a_m / y);
	}
	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) <= -50000000000000.0:
		tmp = y * (x / a_m)
	elif (x * y) <= 2e-23:
		tmp = z * (t / -a_m)
	else:
		tmp = x / (a_m / y)
	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) <= -50000000000000.0)
		tmp = Float64(y * Float64(x / a_m));
	elseif (Float64(x * y) <= 2e-23)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	else
		tmp = Float64(x / Float64(a_m / y));
	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) <= -50000000000000.0)
		tmp = y * (x / a_m);
	elseif ((x * y) <= 2e-23)
		tmp = z * (t / -a_m);
	else
		tmp = x / (a_m / y);
	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], -50000000000000.0], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-23], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e13

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

   if -5e13 < (*.f64 x y) < 1.99999999999999992e-23

   1. Initial program 90.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 74.2

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

   5. Simplified 74.2%

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

      1. distribute-rgt-neg-out N/A

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

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

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

      3. associate-*l/ N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 77.7

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 77.7

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

   7. Applied egg-rr 77.7%

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

   if 1.99999999999999992e-23 < (*.f64 x y)

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.6

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

   5. Simplified 79.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.3

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

   7. Applied egg-rr 74.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, -\frac{z \cdot t}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z}{a\_m}, \frac{x \cdot y}{a\_m}\right)\\ \end{array} \end{array} \]

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 (<= t 7.2e-50)
    (fma (/ y a_m) x (- (/ (* z t) a_m)))
    (fma (- t) (/ z a_m) (/ (* x y) a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= 7.2e-50) {
		tmp = fma((y / a_m), x, -((z * t) / a_m));
	} else {
		tmp = fma(-t, (z / a_m), ((x * y) / a_m));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (t <= 7.2e-50)
		tmp = fma(Float64(y / a_m), x, Float64(-Float64(Float64(z * t) / a_m)));
	else
		tmp = fma(Float64(-t), Float64(z / a_m), Float64(Float64(x * y) / 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[t, 7.2e-50], N[(N[(y / a$95$m), $MachinePrecision] * x + (-N[(N[(z * t), $MachinePrecision] / a$95$m), $MachinePrecision])), $MachinePrecision], N[((-t) * N[(z / a$95$m), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.19999999999999958e-50

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 91.0

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

   4. Applied egg-rr 91.0%

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

   if 7.19999999999999958e-50 < t

   1. Initial program 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 92.6

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

   4. Applied egg-rr 92.6%

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

Alternative 7: 92.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{t}{a\_m}, \frac{x \cdot y}{a\_m}\right)\\ \end{array} \end{array} \]

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 (<= a_m 5e-114)
    (/ (- (* x y) (* z t)) a_m)
    (fma (- z) (/ t a_m) (/ (* x y) a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 5e-114) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma(-z, (t / a_m), ((x * y) / a_m));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 5e-114)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(-z), Float64(t / a_m), Float64(Float64(x * y) / 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, 5e-114], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[((-z) * N[(t / a$95$m), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999989e-114

   1. Initial program 90.3%

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

   if 4.99999999999999989e-114 < a

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 97.4

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

   4. Applied egg-rr 97.4%

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

Alternative 8: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \frac{t}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \end{array} \end{array} \]

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) -50000000000000.0)
    (* y (/ x a_m))
    (if (<= (* x y) 2e-23) (* z (/ t (- a_m))) (* x (/ y a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = z * (t / -a_m);
	} else {
		tmp = x * (y / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-50000000000000.0d0)) then
        tmp = y * (x / a_m)
    else if ((x * y) <= 2d-23) then
        tmp = z * (t / -a_m)
    else
        tmp = x * (y / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = z * (t / -a_m);
	} else {
		tmp = x * (y / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -50000000000000.0:
		tmp = y * (x / a_m)
	elif (x * y) <= 2e-23:
		tmp = z * (t / -a_m)
	else:
		tmp = x * (y / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -50000000000000.0)
		tmp = Float64(y * Float64(x / a_m));
	elseif (Float64(x * y) <= 2e-23)
		tmp = Float64(z * Float64(t / Float64(-a_m)));
	else
		tmp = Float64(x * Float64(y / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -50000000000000.0)
		tmp = y * (x / a_m);
	elseif ((x * y) <= 2e-23)
		tmp = z * (t / -a_m);
	else
		tmp = x * (y / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -50000000000000.0], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-23], N[(z * N[(t / (-a$95$m)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e13

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

   if -5e13 < (*.f64 x y) < 1.99999999999999992e-23

   1. Initial program 90.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 74.2

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

   5. Simplified 74.2%

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

      1. distribute-rgt-neg-out N/A

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

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

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

      3. associate-*l/ N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 77.7

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 77.7

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

   7. Applied egg-rr 77.7%

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

   if 1.99999999999999992e-23 < (*.f64 x y)

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.6

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

   5. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 75.3

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

   7. Applied egg-rr 75.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \end{array} \end{array} \]

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) -50000000000000.0)
    (* y (/ x a_m))
    (if (<= (* x y) 2e-23) (* t (/ z (- a_m))) (* x (/ y a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = t * (z / -a_m);
	} else {
		tmp = x * (y / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: tmp
    if ((x * y) <= (-50000000000000.0d0)) then
        tmp = y * (x / a_m)
    else if ((x * y) <= 2d-23) then
        tmp = t * (z / -a_m)
    else
        tmp = x * (y / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -50000000000000.0) {
		tmp = y * (x / a_m);
	} else if ((x * y) <= 2e-23) {
		tmp = t * (z / -a_m);
	} else {
		tmp = x * (y / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (x * y) <= -50000000000000.0:
		tmp = y * (x / a_m)
	elif (x * y) <= 2e-23:
		tmp = t * (z / -a_m)
	else:
		tmp = x * (y / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(x * y) <= -50000000000000.0)
		tmp = Float64(y * Float64(x / a_m));
	elseif (Float64(x * y) <= 2e-23)
		tmp = Float64(t * Float64(z / Float64(-a_m)));
	else
		tmp = Float64(x * Float64(y / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((x * y) <= -50000000000000.0)
		tmp = y * (x / a_m);
	elseif ((x * y) <= 2e-23)
		tmp = t * (z / -a_m);
	else
		tmp = x * (y / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(x * y), $MachinePrecision], -50000000000000.0], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-23], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e13

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

   if -5e13 < (*.f64 x y) < 1.99999999999999992e-23

   1. Initial program 90.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lower-*.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. lower-neg.f64 74.2

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

   5. Simplified 74.2%

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

      1. lift-neg.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 76.8

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

   7. Applied egg-rr 76.8%

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

   if 1.99999999999999992e-23 < (*.f64 x y)

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.6

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

   5. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   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. lower-*.f64 N/A

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

      4. lower-/.f64 75.3

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

   7. Applied egg-rr 75.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000000:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.7% 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 90.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 56.9

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

5. Simplified 56.9%

   \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
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. lower-*.f64 N/A

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

   4. lower-/.f64 57.9

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

7. Applied egg-rr 57.9%

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

8. Final simplification 57.9%

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

Alternative 11: 51.6% 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 90.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 56.9

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

5. Simplified 56.9%

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

   1. associate-*l/ N/A

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

   2. lift-/.f64 N/A

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

   3. lower-*.f64 58.8

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

7. Applied egg-rr 58.8%

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

8. Final simplification 58.8%

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

Developer Target 1: 92.1% 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 2024207 
(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))