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

Percentage Accurate: 91.7% → 95.4%

Time: 9.3s

Alternatives: 8

Speedup: 0.7×

• 27.8% 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: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 100000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, \frac{-z}{a\_m} \cdot t\right)\\ \end{array} \end{array} \]

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 100000000000.0)
    (/ (fma (- z) t (* x y)) a_m)
    (fma (/ x a_m) y (* (/ (- 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 <= 100000000000.0) {
		tmp = fma(-z, t, (x * y)) / a_m;
	} else {
		tmp = fma((x / a_m), y, ((-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 <= 100000000000.0)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a_m);
	else
		tmp = fma(Float64(x / a_m), y, Float64(Float64(Float64(-z) / a_m) * 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, 100000000000.0], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y + N[(N[((-z) / a$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1e11

   1. Initial program 95.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 95.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

   if 1e11 < a

   1. Initial program 82.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      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. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 100000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{-z}{a} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-t}{\frac{a\_m}{z}}\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a\_m} \cdot t\\ \end{array} \end{array} \]

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 z) -5e+51)
    (/ (- t) (/ a_m z))
    (if (<= (* t z) 2e-63) (/ (* x y) a_m) (* (/ (- 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 ((t * z) <= -5e+51) {
		tmp = -t / (a_m / z);
	} else if ((t * z) <= 2e-63) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (-z / a_m) * t;
	}
	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 ((t * z) <= (-5d+51)) then
        tmp = -t / (a_m / z)
    else if ((t * z) <= 2d-63) then
        tmp = (x * y) / a_m
    else
        tmp = (-z / a_m) * t
    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 ((t * z) <= -5e+51) {
		tmp = -t / (a_m / z);
	} else if ((t * z) <= 2e-63) {
		tmp = (x * y) / a_m;
	} else {
		tmp = (-z / a_m) * t;
	}
	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 (t * z) <= -5e+51:
		tmp = -t / (a_m / z)
	elif (t * z) <= 2e-63:
		tmp = (x * y) / a_m
	else:
		tmp = (-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 (Float64(t * z) <= -5e+51)
		tmp = Float64(Float64(-t) / Float64(a_m / z));
	elseif (Float64(t * z) <= 2e-63)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = Float64(Float64(Float64(-z) / a_m) * t);
	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 ((t * z) <= -5e+51)
		tmp = -t / (a_m / z);
	elseif ((t * z) <= 2e-63)
		tmp = (x * y) / a_m;
	else
		tmp = (-z / a_m) * t;
	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[(t * z), $MachinePrecision], -5e+51], N[((-t) / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2e-63], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[((-z) / a$95$m), $MachinePrecision] * t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e51

   1. Initial program 82.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 82.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-/r* N/A

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

      12. frac-2neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 N/A

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

      20. distribute-frac-neg2 N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 81.8

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

   9. Applied rewrites 81.8%

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

   11. Applied rewrites 81.8%

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

   if -5e51 < (*.f64 z t) < 2.00000000000000013e-63

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if 2.00000000000000013e-63 < (*.f64 z t)

   1. Initial program 89.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{-z}{a\_m} \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (/ (- z) a_m) t)))
   (*
    a_s
    (if (<= (* t z) -5e+51) t_1 (if (<= (* t z) 2e-63) (/ (* 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 = (-z / a_m) * t;
	double tmp;
	if ((t * z) <= -5e+51) {
		tmp = t_1;
	} else if ((t * z) <= 2e-63) {
		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 = (-z / a_m) * t
    if ((t * z) <= (-5d+51)) then
        tmp = t_1
    else if ((t * z) <= 2d-63) 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 = (-z / a_m) * t;
	double tmp;
	if ((t * z) <= -5e+51) {
		tmp = t_1;
	} else if ((t * z) <= 2e-63) {
		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 = (-z / a_m) * t
	tmp = 0
	if (t * z) <= -5e+51:
		tmp = t_1
	elif (t * z) <= 2e-63:
		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(Float64(Float64(-z) / a_m) * t)
	tmp = 0.0
	if (Float64(t * z) <= -5e+51)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e-63)
		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 = (-z / a_m) * t;
	tmp = 0.0;
	if ((t * z) <= -5e+51)
		tmp = t_1;
	elseif ((t * z) <= 2e-63)
		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[(N[((-z) / a$95$m), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], -5e+51], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e-63], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e51 or 2.00000000000000013e-63 < (*.f64 z t)

   1. Initial program 86.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -5e51 < (*.f64 z t) < 2.00000000000000013e-63

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\frac{-t}{\frac{a\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a\_m}\\ \end{array} \end{array} \]

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 (<= (* t z) -2e+259) (/ (- t) (/ a_m z)) (/ (fma (- z) t (* 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 * z) <= -2e+259) {
		tmp = -t / (a_m / z);
	} else {
		tmp = fma(-z, t, (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(t * z) <= -2e+259)
		tmp = Float64(Float64(-t) / Float64(a_m / z));
	else
		tmp = Float64(fma(Float64(-z), t, 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[(t * z), $MachinePrecision], -2e+259], N[((-t) / N[(a$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e259

   1. Initial program 62.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 62.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.1

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

   4. Applied rewrites 62.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-/r* N/A

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

      12. frac-2neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 N/A

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

      20. distribute-frac-neg2 N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

   if -2e259 < (*.f64 z t)

   1. Initial program 94.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 95.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 95.3

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

   4. Applied rewrites 95.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\frac{-t}{\frac{a\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a\_m}\\ \end{array} \end{array} \]

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 (<= (* t z) -2e+259) (/ (- t) (/ a_m z)) (/ (- (* 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 ((t * z) <= -2e+259) {
		tmp = -t / (a_m / z);
	} else {
		tmp = ((x * y) - (t * z)) / 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 ((t * z) <= (-2d+259)) then
        tmp = -t / (a_m / z)
    else
        tmp = ((x * y) - (t * z)) / 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 ((t * z) <= -2e+259) {
		tmp = -t / (a_m / z);
	} 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 (t * z) <= -2e+259:
		tmp = -t / (a_m / z)
	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(t * z) <= -2e+259)
		tmp = Float64(Float64(-t) / Float64(a_m / z));
	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 ((t * z) <= -2e+259)
		tmp = -t / (a_m / z);
	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[(t * z), $MachinePrecision], -2e+259], N[((-t) / N[(a$95$m / z), $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 z t) < -2e259

   1. Initial program 62.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 62.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.1

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

   4. Applied rewrites 62.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-/r* N/A

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

      12. frac-2neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 N/A

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

      20. distribute-frac-neg2 N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

   if -2e259 < (*.f64 z t)

   1. Initial program 94.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y - t \cdot z \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \end{array} \end{array} \]

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) (* t z)) -2e+278) (* (/ x a_m) y) (/ (* 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) - (t * z)) <= -2e+278) {
		tmp = (x / a_m) * y;
	} 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) - (t * z)) <= (-2d+278)) then
        tmp = (x / a_m) * y
    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) - (t * z)) <= -2e+278) {
		tmp = (x / a_m) * y;
	} 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) - (t * z)) <= -2e+278:
		tmp = (x / a_m) * y
	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(Float64(x * y) - Float64(t * z)) <= -2e+278)
		tmp = Float64(Float64(x / a_m) * y);
	else
		tmp = Float64(Float64(x * y) / a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (((x * y) - (t * z)) <= -2e+278)
		tmp = (x / a_m) * y;
	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[(N[(x * y), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision], -2e+278], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.99999999999999993e278

   1. Initial program 73.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -1.99999999999999993e278 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 94.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - t \cdot z \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-258}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a\_m} \cdot y\\ \end{array} \end{array} \]

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 (<= z -7.2e-258) (* (/ y a_m) x) (* (/ 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 (z <= -7.2e-258) {
		tmp = (y / a_m) * x;
	} 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 (z <= (-7.2d-258)) then
        tmp = (y / a_m) * x
    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 (z <= -7.2e-258) {
		tmp = (y / a_m) * x;
	} 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 z <= -7.2e-258:
		tmp = (y / a_m) * x
	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 (z <= -7.2e-258)
		tmp = Float64(Float64(y / a_m) * x);
	else
		tmp = Float64(Float64(x / 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 (z <= -7.2e-258)
		tmp = (y / a_m) * x;
	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[z, -7.2e-258], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.19999999999999958e-258

   1. Initial program 93.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.4

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

   8. Applied rewrites 51.4%

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

   10. Applied rewrites 49.6%

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

   if -7.19999999999999958e-258 < z

   1. Initial program 90.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1

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

   8. Applied rewrites 51.1%

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

Alternative 8: 51.7% accurate, 1.5× speedup

Math

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

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 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) {
	return a_s * ((x / a_m) * y);
}

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 / a_m) * y)
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 / a_m) * y);
}

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 / a_m) * y)

Julia

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.2

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in t around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 51.3

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

8. Applied rewrites 51.3%

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

Developer Target 1: 92.3% 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 2024267 
(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))