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

Percentage Accurate: 91.2% → 95.2%

Time: 10.4s

Alternatives: 6

Speedup: 0.7×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.2% 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: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 10^{-13}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, 0 - z \cdot \frac{t}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 1e-13)
    (/ (- (* x y) (* z t)) a_m)
    (fma (/ x a_m) y (- 0.0 (* 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1e-13) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = fma((x / a_m), y, (0.0 - (z * (t / a_m))));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1e-13)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = fma(Float64(x / a_m), y, Float64(0.0 - Float64(z * Float64(t / a_m))));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 1e-13], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x / a$95$m), $MachinePrecision] * y + N[(0.0 - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1e-13

   1. Initial program 95.1%

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

   if 1e-13 < a

   1. Initial program 83.5%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{a}, y, \color{blue}{0 - \frac{z \cdot t}{a}}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x}{a}, y, \color{blue}{0 - \frac{z \cdot t}{a}}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{a}, y, 0 - \color{blue}{\frac{z \cdot t}{a}}\right) \]

      11. *-lowering-*.f64 88.2

          \[\leadsto \mathsf{fma}\left(\frac{x}{a}, y, 0 - \frac{\color{blue}{z \cdot t}}{a}\right) \]

   4. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, y, 0 - \frac{z \cdot t}{a}\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 91.0

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

   6. Applied egg-rr 91.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{-13}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, 0 - z \cdot \frac{t}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := 0 - z \cdot \frac{t}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- 0.0 (* z (/ t a_m)))))
   (*
    a_s
    (if (<= (* z t) -5e+23) t_1 (if (<= (* z t) 2e+43) (* 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = 0.0 - (z * (t / a_m));
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e+43) {
		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.
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 = 0.0d0 - (z * (t / a_m))
    if ((z * t) <= (-5d+23)) then
        tmp = t_1
    else if ((z * t) <= 2d+43) 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;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = 0.0 - (z * (t / a_m));
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e+43) {
		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])
def code(a_s, x, y, z, t, a_m):
	t_1 = 0.0 - (z * (t / a_m))
	tmp = 0
	if (z * t) <= -5e+23:
		tmp = t_1
	elif (z * t) <= 2e+43:
		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])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(0.0 - Float64(z * Float64(t / a_m)))
	tmp = 0.0
	if (Float64(z * t) <= -5e+23)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+43)
		tmp = Float64(x * Float64(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])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = 0.0 - (z * (t / a_m));
	tmp = 0.0;
	if ((z * t) <= -5e+23)
		tmp = t_1;
	elseif ((z * t) <= 2e+43)
		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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(0.0 - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -5e+23], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+43], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 2.00000000000000003e43 < (*.f64 z t)

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -1 \cdot \frac{t \cdot z}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + 0} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot z}{a}\right)\right)} + 0 \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z}{a}}\right)\right) + 0 \]

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

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(\frac{z}{a}\right)\right)} + 0 \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \frac{z}{a}\right)} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \frac{z}{a}, 0\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(\frac{z}{a}\right)}, 0\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      11. /-lowering-/.f64 81.5

          \[\leadsto \mathsf{fma}\left(t, 0 - \color{blue}{\frac{z}{a}}, 0\right) \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, 0 - \frac{z}{a}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(0 - \frac{z}{a}\right)} \]

      2. sub0-neg N/A

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

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

      12. /-lowering-/.f64 80.7

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

   7. Applied egg-rr 80.7%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 2.00000000000000003e43

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 75.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 76.5

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

   7. Applied egg-rr 76.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;0 - z \cdot \frac{t}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := 0 - t \cdot \frac{z}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- 0.0 (* t (/ z a_m)))))
   (*
    a_s
    (if (<= (* z t) -5e+23) t_1 (if (<= (* z t) 2e+43) (* 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = 0.0 - (t * (z / a_m));
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e+43) {
		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.
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 = 0.0d0 - (t * (z / a_m))
    if ((z * t) <= (-5d+23)) then
        tmp = t_1
    else if ((z * t) <= 2d+43) 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;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = 0.0 - (t * (z / a_m));
	double tmp;
	if ((z * t) <= -5e+23) {
		tmp = t_1;
	} else if ((z * t) <= 2e+43) {
		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])
def code(a_s, x, y, z, t, a_m):
	t_1 = 0.0 - (t * (z / a_m))
	tmp = 0
	if (z * t) <= -5e+23:
		tmp = t_1
	elif (z * t) <= 2e+43:
		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])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(0.0 - Float64(t * Float64(z / a_m)))
	tmp = 0.0
	if (Float64(z * t) <= -5e+23)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+43)
		tmp = Float64(x * Float64(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])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = 0.0 - (t * (z / a_m));
	tmp = 0.0;
	if ((z * t) <= -5e+23)
		tmp = t_1;
	elseif ((z * t) <= 2e+43)
		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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(0.0 - N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], -5e+23], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+43], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 2.00000000000000003e43 < (*.f64 z t)

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -1 \cdot \frac{t \cdot z}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + 0} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot z}{a}\right)\right)} + 0 \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z}{a}}\right)\right) + 0 \]

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

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(\frac{z}{a}\right)\right)} + 0 \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \frac{z}{a}\right)} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \frac{z}{a}, 0\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(\frac{z}{a}\right)}, 0\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      11. /-lowering-/.f64 81.5

          \[\leadsto \mathsf{fma}\left(t, 0 - \color{blue}{\frac{z}{a}}, 0\right) \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, 0 - \frac{z}{a}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(0 - \frac{z}{a}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(0 - \frac{z}{a}\right) \cdot t} \]

      3. sub0-neg N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 81.5

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

   7. Applied egg-rr 81.5%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 2.00000000000000003e43

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 75.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 76.5

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

   7. Applied egg-rr 76.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23}:\\ \;\;\;\;0 - t \cdot \frac{z}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \frac{t}{a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* z t) 4e+302)
    (/ (- (* x y) (* z t)) a_m)
    (- 0.0 (* 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= 4e+302) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = 0.0 - (z * (t / a_m));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 * t) <= 4d+302) then
        tmp = ((x * y) - (z * t)) / a_m
    else
        tmp = 0.0d0 - (z * (t / a_m))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= 4e+302) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = 0.0 - (z * (t / a_m));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if (z * t) <= 4e+302:
		tmp = ((x * y) - (z * t)) / a_m
	else:
		tmp = 0.0 - (z * (t / a_m))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(z * t) <= 4e+302)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = Float64(0.0 - Float64(z * Float64(t / a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if ((z * t) <= 4e+302)
		tmp = ((x * y) - (z * t)) / a_m;
	else
		tmp = 0.0 - (z * (t / a_m));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], 4e+302], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(0.0 - N[(z * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.0000000000000003e302

   1. Initial program 94.5%

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

   if 4.0000000000000003e302 < (*.f64 z t)

   1. Initial program 65.6%

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

   3. Taylor expanded in x around 0

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{0 + -1 \cdot \frac{t \cdot z}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + 0} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot z}{a}\right)\right)} + 0 \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z}{a}}\right)\right) + 0 \]

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

         \[\leadsto \color{blue}{t \cdot \left(\mathsf{neg}\left(\frac{z}{a}\right)\right)} + 0 \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \frac{z}{a}\right)} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, -1 \cdot \frac{z}{a}, 0\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(\frac{z}{a}\right)}, 0\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{0 - \frac{z}{a}}, 0\right) \]

      11. /-lowering-/.f64 96.0

          \[\leadsto \mathsf{fma}\left(t, 0 - \color{blue}{\frac{z}{a}}, 0\right) \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, 0 - \frac{z}{a}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot \left(0 - \frac{z}{a}\right)} \]

      2. sub0-neg N/A

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

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

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

      12. /-lowering-/.f64 95.8

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

   7. Applied egg-rr 95.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \frac{y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

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.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (* a_s (if (<= t 2.2e+30) (* x (/ y 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= 2.2e+30) {
		tmp = x * (y / a_m);
	} else {
		tmp = y * (x / 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.
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 <= 2.2d+30) then
        tmp = x * (y / a_m)
    else
        tmp = y * (x / 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;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (t <= 2.2e+30) {
		tmp = x * (y / a_m);
	} else {
		tmp = y * (x / 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])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if t <= 2.2e+30:
		tmp = x * (y / a_m)
	else:
		tmp = 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])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (t <= 2.2e+30)
		tmp = Float64(x * Float64(y / a_m));
	else
		tmp = Float64(y * Float64(x / 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])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (t <= 2.2e+30)
		tmp = x * (y / a_m);
	else
		tmp = y * (x / 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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[t, 2.2e+30], N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2e30

   1. Initial program 92.4%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 51.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 55.3

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

   7. Applied egg-rr 55.3%

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

   if 2.2e30 < t

   1. Initial program 89.0%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

      3. accelerator-lowering-fma.f64 37.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

   5. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

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

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

      4. /-lowering-/.f64 39.5

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

   7. Applied egg-rr 39.5%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 1.5× speedup

Math

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

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(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);
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.
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;
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])
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])
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(x * Float64(y / a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in x around inf

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

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

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

   2. +-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{x \cdot y + 0}}{a} \]

   3. accelerator-lowering-fma.f64 48.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, 0\right)}}{a} \]

5. Simplified 48.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, 0\right)}{a}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 51.7

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

7. Applied egg-rr 51.7%

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

Developer Target 1: 92.0% 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 2024195 
(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))