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

Percentage Accurate: 91.7% → 93.8%

Time: 11.5s

Alternatives: 10

Speedup: 1.0×

• 28.1% 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: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, \frac{z}{\frac{a\_m}{-t}}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 1.6e+101)
    (/ (- (* x y) (* z t)) a_m)
    (fma (/ y a_m) x (/ z (/ a_m (- t)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1.6e+101) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma((y / a_m), x, (z / (a_m / -t)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1.6e+101)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(y / a_m), x, Float64(z / Float64(a_m / Float64(-t))));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 1.6e+101], 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[(a$95$m / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.60000000000000003e101

   1. Initial program 95.3%

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

   if 1.60000000000000003e101 < a

   1. Initial program 80.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. 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 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

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

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

      6. lift-neg.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{z}{\frac{a}{-t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z}{a\_m}, \frac{x}{\frac{a\_m}{y}}\right)\\ \end{array} \end{array} \]

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 9.8e+120)
    (/ (- (* x y) (* z t)) a_m)
    (fma (- t) (/ z 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 (a_m <= 9.8e+120) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma(-t, (z / a_m), (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 (a_m <= 9.8e+120)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(-t), Float64(z / a_m), Float64(x / Float64(a_m / y)));
	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, 9.8e+120], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[((-t) * N[(z / a$95$m), $MachinePrecision] + N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.80000000000000021e120

   1. Initial program 95.4%

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

   if 9.80000000000000021e120 < a

   1. Initial program 78.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. +-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 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.1

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

   6. Applied rewrites 88.1%

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

Alternative 3: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := -z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* z t))))
   (*
    a_s
    (if (<= (* z t) -4e+31)
      (/ t_1 a_m)
      (if (<= (* z t) 5e-122) (/ (* x y) a_m) (* t_1 (/ 1.0 a_m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = -(z * t);
	double tmp;
	if ((z * t) <= -4e+31) {
		tmp = t_1 / a_m;
	} else if ((z * t) <= 5e-122) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1 * (1.0 / a_m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(z * t)
    if ((z * t) <= (-4d+31)) then
        tmp = t_1 / a_m
    else if ((z * t) <= 5d-122) then
        tmp = (x * y) / a_m
    else
        tmp = t_1 * (1.0d0 / a_m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = -(z * t);
	double tmp;
	if ((z * t) <= -4e+31) {
		tmp = t_1 / a_m;
	} else if ((z * t) <= 5e-122) {
		tmp = (x * y) / a_m;
	} else {
		tmp = t_1 * (1.0 / a_m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = -(z * t)
	tmp = 0
	if (z * t) <= -4e+31:
		tmp = t_1 / a_m
	elif (z * t) <= 5e-122:
		tmp = (x * y) / a_m
	else:
		tmp = t_1 * (1.0 / a_m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(-Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -4e+31)
		tmp = Float64(t_1 / a_m);
	elseif (Float64(z * t) <= 5e-122)
		tmp = Float64(Float64(x * y) / a_m);
	else
		tmp = Float64(t_1 * Float64(1.0 / a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = -(z * t);
	tmp = 0.0;
	if ((z * t) <= -4e+31)
		tmp = t_1 / a_m;
	elseif ((z * t) <= 5e-122)
		tmp = (x * y) / a_m;
	else
		tmp = t_1 * (1.0 / a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = (-N[(z * t), $MachinePrecision])}, N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -4e+31], N[(t$95$1 / a$95$m), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-122], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], N[(t$95$1 * N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -3.9999999999999999e31

   1. Initial program 92.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 76.8

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

   5. Applied rewrites 76.8%

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

   if -3.9999999999999999e31 < (*.f64 z t) < 4.9999999999999999e-122

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 4.9999999999999999e-122 < (*.f64 z t)

   1. Initial program 89.8%

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

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.1

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

   7. Applied rewrites 66.1%

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\left(t \cdot \left(-z\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z \cdot t\right) \cdot \frac{1}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{a\_m}{t}}{z}}\\ \end{array} \end{array} \]

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) 2e+296)
    (/ (fma (- z) t (* x y)) a_m)
    (/ -1.0 (/ (/ a_m t) z)))))

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) <= 2e+296) {
		tmp = fma(-z, t, (x * y)) / a_m;
	} else {
		tmp = -1.0 / ((a_m / t) / z);
	}
	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) <= 2e+296)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a_m);
	else
		tmp = Float64(-1.0 / Float64(Float64(a_m / t) / z));
	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[(z * t), $MachinePrecision], 2e+296], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(-1.0 / N[(N[(a$95$m / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.99999999999999996e296

   1. Initial program 94.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 94.3

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

   4. Applied rewrites 94.3%

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

   if 1.99999999999999996e296 < (*.f64 z t)

   1. Initial program 63.4%

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

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 63.3

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

   7. Applied rewrites 63.3%

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

   9. Applied rewrites 99.9%

      \[\leadsto \frac{-1}{\frac{\frac{a}{t}}{\color{blue}{z}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 3.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z}{a\_m}, y \cdot \frac{x}{a\_m}\right)\\ \end{array} \end{array} \]

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 3.8e+121)
    (/ (- (* x y) (* z t)) a_m)
    (fma (- t) (/ z a_m) (* y (/ x a_m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 3.8e+121) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma(-t, (z / a_m), (y * (x / a_m)));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 3.8e+121)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(-t), Float64(z / a_m), Float64(y * Float64(x / 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, 3.8e+121], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[((-t) * N[(z / a$95$m), $MachinePrecision] + N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 3.8e121

   1. Initial program 95.4%

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

   if 3.8e121 < a

   1. Initial program 78.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. +-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 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.0

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

   6. Applied rewrites 92.0%

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

Alternative 6: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{-z \cdot t}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 t)) a_m)))
   (*
    a_s
    (if (<= (* z t) -4e+31)
      t_1
      (if (<= (* z t) 5e-122) (/ (* 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 * t) / a_m;
	double tmp;
	if ((z * t) <= -4e+31) {
		tmp = t_1;
	} else if ((z * t) <= 5e-122) {
		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 * t) / a_m
    if ((z * t) <= (-4d+31)) then
        tmp = t_1
    else if ((z * t) <= 5d-122) 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 * t) / a_m;
	double tmp;
	if ((z * t) <= -4e+31) {
		tmp = t_1;
	} else if ((z * t) <= 5e-122) {
		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 * t) / a_m
	tmp = 0
	if (z * t) <= -4e+31:
		tmp = t_1
	elif (z * t) <= 5e-122:
		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 * t)) / a_m)
	tmp = 0.0
	if (Float64(z * t) <= -4e+31)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-122)
		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 * t) / a_m;
	tmp = 0.0;
	if ((z * t) <= -4e+31)
		tmp = t_1;
	elseif ((z * t) <= 5e-122)
		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 * t), $MachinePrecision]) / a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -4e+31], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-122], 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) < -3.9999999999999999e31 or 4.9999999999999999e-122 < (*.f64 z t)

   1. Initial program 91.3%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   if -3.9999999999999999e31 < (*.f64 z t) < 4.9999999999999999e-122

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 1.0× speedup

Math

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

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 (/ (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) {
	return a_s * (fma(-z, t, (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(fma(Float64(-z), t, Float64(x * y)) / a_m))
end

Wolfram

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

Derivation

1. Initial program 92.8%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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 92.8

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

4. Applied rewrites 92.8%

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

Alternative 8: 91.7% accurate, 1.0× speedup

Math

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

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) (* 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) {
	return a_s * (((x * y) - (z * t)) / 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) - (z * t)) / 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) - (z * t)) / 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) - (z * t)) / a_m)

Julia

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

Alternative 9: 51.1% accurate, 1.5× speedup

Math

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

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(Float64(x * y) / a_m))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 51.9

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

5. Applied rewrites 51.9%

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

Alternative 10: 8.9% accurate, 1.5× speedup

Math

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

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

Derivation

1. Initial program 92.8%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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 92.8

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

4. Applied rewrites 92.8%

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

6. Applied rewrites 49.3%

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

7. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 10.0

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

9. Applied rewrites 10.0%

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

11. Applied rewrites 10.8%

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

12. Final simplification 10.8%

    \[\leadsto z \cdot \frac{t}{a} \]
13. Add Preprocessing

Developer Target 1: 91.7% 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 2024232 
(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))