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

Percentage Accurate: 91.0% → 95.6%

Time: 11.0s

Alternatives: 11

Speedup: 0.6×

• 28.4% 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: 91.0% 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: 95.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{a\_m}}, \frac{x}{\sqrt{a\_m}}, \left(-t\right) \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 1.2e+14)
    (/ (fma x y (* z (- t))) a_m)
    (fma (/ y (sqrt a_m)) (/ x (sqrt a_m)) (* (- t) (/ z a_m))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 1.2e+14) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = fma((y / sqrt(a_m)), (x / sqrt(a_m)), (-t * (z / a_m)));
	}
	return a_s * tmp;
}

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 1.2e+14)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = fma(Float64(y / sqrt(a_m)), Float64(x / sqrt(a_m)), Float64(Float64(-t) * Float64(z / 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, 1.2e+14], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision] + N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.2e14

   1. Initial program 92.0%

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

      1. fma-neg 92.5%

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

      2. distribute-rgt-neg-out 92.5%

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

   3. Simplified 92.5%

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

   if 1.2e14 < a

   1. Initial program 88.6%

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

      1. div-sub 88.6%

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

      2. *-commutative 88.6%

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

      3. add-sqr-sqrt 88.4%

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

      4. times-frac 91.4%

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

      5. fma-neg 91.4%

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

      6. *-commutative 91.4%

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

      7. associate-/l* 95.0%

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

   4. Applied egg-rr 95.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{a}}, \frac{x}{\sqrt{a}}, \left(-t\right) \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a\_m}, \left(-t\right) \cdot \frac{z}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 9.2e-35)
    (/ (fma x y (* z (- t))) a_m)
    (fma y (/ x a_m) (* (- t) (/ z a_m))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 9.2e-35) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = fma(y, (x / a_m), (-t * (z / a_m)));
	}
	return a_s * tmp;
}

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 9.2e-35)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = fma(y, Float64(x / a_m), Float64(Float64(-t) * Float64(z / 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, 9.2e-35], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision] + N[((-t) * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.1999999999999996e-35

   1. Initial program 91.7%

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

      1. fma-neg 92.2%

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

      2. distribute-rgt-neg-out 92.2%

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

   3. Simplified 92.2%

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

   if 9.1999999999999996e-35 < a

   1. Initial program 89.8%

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

      1. div-sub 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-/l* 89.6%

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

      4. fma-neg 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-/l* 92.9%

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

   4. Applied egg-rr 92.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \left(-t\right) \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a\_m}, \frac{t}{\frac{a\_m}{-z}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 5e-35)
    (/ (fma x y (* z (- t))) a_m)
    (fma y (/ x a_m) (/ t (/ a_m (- z)))))))

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 <= 5e-35) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = fma(y, (x / a_m), (t / (a_m / -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])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 5e-35)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = fma(y, Float64(x / a_m), Float64(t / Float64(a_m / Float64(-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.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 5e-35], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision] + N[(t / N[(a$95$m / (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999964e-35

   1. Initial program 91.7%

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

      1. fma-neg 92.2%

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

      2. distribute-rgt-neg-out 92.2%

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

   3. Simplified 92.2%

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

   if 4.99999999999999964e-35 < a

   1. Initial program 89.8%

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

      1. div-sub 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-/l* 89.6%

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

      4. fma-neg 89.6%

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

      5. *-commutative 89.6%

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

      6. associate-/l* 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. clear-num 92.5%

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

      2. un-div-inv 92.5%

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

   6. Applied egg-rr 92.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{t}{\frac{a}{-z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m} - t \cdot \frac{z}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 4.6e-23)
    (/ (fma x y (* z (- t))) a_m)
    (- (* x (/ y a_m)) (* t (/ z a_m))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 4.6e-23) {
		tmp = fma(x, y, (z * -t)) / a_m;
	} else {
		tmp = (x * (y / a_m)) - (t * (z / a_m));
	}
	return a_s * tmp;
}

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 4.6e-23)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a_m);
	else
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(t * Float64(z / 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, 4.6e-23], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 4.6000000000000002e-23

   1. Initial program 91.8%

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

      1. fma-neg 92.3%

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

      2. distribute-rgt-neg-out 92.3%

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

   3. Simplified 92.3%

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

   if 4.6000000000000002e-23 < a

   1. Initial program 89.4%

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

      1. div-sub 89.4%

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

      2. associate-/l* 90.8%

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

      3. *-commutative 90.8%

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

      4. associate-/l* 94.1%

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

   4. Applied egg-rr 94.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-52}:\\ \;\;\;\;t \cdot \frac{z}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 (<= (* x y) -1e+28)
    (/ x (/ a_m y))
    (if (<= (* x y) 5e-52) (* 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -1e+28) {
		tmp = x / (a_m / y);
	} else if ((x * y) <= 5e-52) {
		tmp = t * (z / -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 ((x * y) <= (-1d+28)) then
        tmp = x / (a_m / y)
    else if ((x * y) <= 5d-52) then
        tmp = t * (z / -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 ((x * y) <= -1e+28) {
		tmp = x / (a_m / y);
	} else if ((x * y) <= 5e-52) {
		tmp = t * (z / -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 (x * y) <= -1e+28:
		tmp = x / (a_m / y)
	elif (x * y) <= 5e-52:
		tmp = t * (z / -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 (Float64(x * y) <= -1e+28)
		tmp = Float64(x / Float64(a_m / y));
	elseif (Float64(x * y) <= 5e-52)
		tmp = Float64(t * Float64(z / Float64(-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 ((x * y) <= -1e+28)
		tmp = x / (a_m / y);
	elseif ((x * y) <= 5e-52)
		tmp = t * (z / -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[N[(x * y), $MachinePrecision], -1e+28], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-52], N[(t * N[(z / (-a$95$m)), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.99999999999999958e27

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf 71.3%

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

      1. associate-*r/ 75.9%

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

   5. Simplified 75.9%

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

      1. clear-num 75.8%

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

      2. un-div-inv 76.4%

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

   7. Applied egg-rr 76.4%

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

   if -9.99999999999999958e27 < (*.f64 x y) < 5e-52

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-*r/ 77.0%

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

      3. distribute-lft-neg-in 77.0%

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

   5. Simplified 77.0%

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

   if 5e-52 < (*.f64 x y)

   1. Initial program 87.8%

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

      1. div-sub 83.7%

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

      2. add-cube-cbrt 83.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      3. times-frac 79.4%

         \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      4. fma-neg 80.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right)} \]

      5. pow2 80.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      6. *-commutative 80.8%

         \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\frac{\color{blue}{t \cdot z}}{a}\right) \]

      7. associate-/l* 80.6%

         \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\color{blue}{t \cdot \frac{z}{a}}\right) \]

   4. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -t \cdot \frac{z}{a}\right)} \]

   5. Taylor expanded in x around inf 65.0%

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

      1. associate-*l/ 63.7%

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

      2. *-commutative 63.7%

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

   7. Simplified 63.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-52}:\\ \;\;\;\;t \cdot \frac{z}{-a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \mathbf{elif}\;x \cdot y \leq 0.5:\\ \;\;\;\;\frac{z \cdot t}{-a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 (<= (* x y) -1e+28)
    (/ x (/ a_m y))
    (if (<= (* x y) 0.5) (/ (* z t) (- 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 ((x * y) <= -1e+28) {
		tmp = x / (a_m / y);
	} else if ((x * y) <= 0.5) {
		tmp = (z * t) / -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 ((x * y) <= (-1d+28)) then
        tmp = x / (a_m / y)
    else if ((x * y) <= 0.5d0) then
        tmp = (z * t) / -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 ((x * y) <= -1e+28) {
		tmp = x / (a_m / y);
	} else if ((x * y) <= 0.5) {
		tmp = (z * t) / -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 (x * y) <= -1e+28:
		tmp = x / (a_m / y)
	elif (x * y) <= 0.5:
		tmp = (z * t) / -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 (Float64(x * y) <= -1e+28)
		tmp = Float64(x / Float64(a_m / y));
	elseif (Float64(x * y) <= 0.5)
		tmp = Float64(Float64(z * t) / Float64(-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 ((x * y) <= -1e+28)
		tmp = x / (a_m / y);
	elseif ((x * y) <= 0.5)
		tmp = (z * t) / -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[N[(x * y), $MachinePrecision], -1e+28], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 0.5], N[(N[(z * t), $MachinePrecision] / (-a$95$m)), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.99999999999999958e27

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf 71.3%

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

      1. associate-*r/ 75.9%

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

   5. Simplified 75.9%

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

      1. clear-num 75.8%

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

      2. un-div-inv 76.4%

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

   7. Applied egg-rr 76.4%

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

   if -9.99999999999999958e27 < (*.f64 x y) < 0.5

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. distribute-rgt-neg-in 76.6%

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

   5. Simplified 76.6%

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

   if 0.5 < (*.f64 x y)

   1. Initial program 86.5%

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

      1. div-sub 81.9%

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

      2. add-cube-cbrt 81.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      3. times-frac 78.7%

         \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      4. fma-neg 80.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right)} \]

      5. pow2 80.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      6. *-commutative 80.2%

         \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\frac{\color{blue}{t \cdot z}}{a}\right) \]

      7. associate-/l* 81.5%

         \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\color{blue}{t \cdot \frac{z}{a}}\right) \]

   4. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -t \cdot \frac{z}{a}\right)} \]

   5. Taylor expanded in x around inf 67.1%

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

      1. associate-*l/ 65.6%

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

      2. *-commutative 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y \leq 0.5:\\ \;\;\;\;\frac{z \cdot t}{-a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 0.6× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 (<= (* x y) (- INFINITY)) (/ y (/ a_m x)) (/ (- (* 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);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = y / (a_m / x);
	} else {
		tmp = ((x * y) - (z * t)) / a_m;
	}
	return a_s * tmp;
}

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = y / (a_m / x);
	} else {
		tmp = ((x * y) - (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 (x * y) <= -math.inf:
		tmp = y / (a_m / x)
	else:
		tmp = ((x * y) - (z * t)) / a_m
	return a_s * tmp

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.9%

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

   3. Taylor expanded in x around inf 54.9%

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

      1. associate-*r/ 93.9%

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

   5. Simplified 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l/ 54.9%

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

      3. associate-*r/ 94.0%

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

      4. clear-num 94.1%

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

      5. un-div-inv 94.2%

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

   7. Applied egg-rr 94.2%

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

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

   1. Initial program 93.8%

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

4. Final simplification 93.8%

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

Alternative 8: 94.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a\_m} - t \cdot \frac{z}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 8e-40)
    (/ (- (* x y) (* z t)) a_m)
    (- (* x (/ y a_m)) (* t (/ z a_m))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 8e-40) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = (x * (y / a_m)) - (t * (z / a_m));
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
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 (a_m <= 8d-40) then
        tmp = ((x * y) - (z * t)) / a_m
    else
        tmp = (x * (y / a_m)) - (t * (z / a_m))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 8e-40) {
		tmp = ((x * y) - (z * t)) / a_m;
	} else {
		tmp = (x * (y / a_m)) - (t * (z / a_m));
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	tmp = 0
	if a_m <= 8e-40:
		tmp = ((x * y) - (z * t)) / a_m
	else:
		tmp = (x * (y / a_m)) - (t * (z / a_m))
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 8e-40)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m);
	else
		tmp = Float64(Float64(x * Float64(y / a_m)) - Float64(t * Float64(z / a_m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	tmp = 0.0;
	if (a_m <= 8e-40)
		tmp = ((x * y) - (z * t)) / a_m;
	else
		tmp = (x * (y / a_m)) - (t * (z / a_m));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 8e-40], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(x * N[(y / a$95$m), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 7.9999999999999994e-40

   1. Initial program 91.7%

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

   if 7.9999999999999994e-40 < a

   1. Initial program 89.8%

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

      1. div-sub 89.7%

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

      2. associate-/l* 91.1%

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

      3. *-commutative 91.1%

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

      4. associate-/l* 94.3%

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

   4. Applied egg-rr 94.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - t \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-248}:\\ \;\;\;\;\frac{x \cdot y}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 1.55e-248) (/ (* 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 <= 1.55e-248) {
		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 <= 1.55d-248) 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 <= 1.55e-248) {
		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 <= 1.55e-248:
		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 <= 1.55e-248)
		tmp = Float64(Float64(x * 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 <= 1.55e-248)
		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, 1.55e-248], N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision], N[(y * N[(x / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5500000000000001e-248

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf 55.0%

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

   if 1.5500000000000001e-248 < t

   1. Initial program 87.6%

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

      1. div-sub 86.0%

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

      2. add-cube-cbrt 85.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      3. times-frac 88.9%

         \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      4. fma-neg 88.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right)} \]

      5. pow2 88.9%

         \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      6. *-commutative 88.9%

         \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\frac{\color{blue}{t \cdot z}}{a}\right) \]

      7. associate-/l* 91.7%

         \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\color{blue}{t \cdot \frac{z}{a}}\right) \]

   4. Applied egg-rr 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -t \cdot \frac{z}{a}\right)} \]

   5. Taylor expanded in x around inf 45.0%

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

      1. associate-*l/ 48.7%

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

      2. *-commutative 48.7%

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

   7. Simplified 48.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-248}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 1.8× 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 1 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 91.2%

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

3. Taylor expanded in x around inf 50.1%

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

   1. associate-*r/ 52.9%

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

5. Simplified 52.9%

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

6. Final simplification 52.9%

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

Alternative 11: 51.0% accurate, 1.8× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 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 (* 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) {
	return a_s * (y * (x / a_m));
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    code = a_s * (y * (x / a_m))
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * (y * (x / a_m));
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	return a_s * (y * (x / a_m))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

   1. div-sub 88.4%

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

   2. add-cube-cbrt 87.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

   3. times-frac 87.3%

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

   4. fma-neg 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right)} \]

   5. pow2 88.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{y}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

   6. *-commutative 88.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\frac{\color{blue}{t \cdot z}}{a}\right) \]

   7. associate-/l* 88.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -\color{blue}{t \cdot \frac{z}{a}}\right) \]

4. Applied egg-rr 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{y}{\sqrt[3]{a}}, -t \cdot \frac{z}{a}\right)} \]

5. Taylor expanded in x around inf 50.1%

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

   1. associate-*l/ 50.9%

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

   2. *-commutative 50.9%

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

7. Simplified 50.9%

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

8. Final simplification 50.9%

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

Developer target: 91.6% accurate, 0.4× 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 2024046 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

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