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

Percentage Accurate: 91.3% → 95.2%

Time: 7.5s

Alternatives: 9

Speedup: 0.6×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.3% 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

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, \frac{-z}{a\_m} \cdot t\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= a_m 5e-26)
    (/ (fma (- z) t (* x y)) a_m)
    (fma (/ y a_m) x (* (/ (- z) a_m) t)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (a_m <= 5e-26) {
		tmp = fma(-z, t, (x * y)) / a_m;
	} else {
		tmp = fma((y / a_m), x, ((-z / a_m) * t));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (a_m <= 5e-26)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a_m);
	else
		tmp = fma(Float64(y / a_m), x, Float64(Float64(Float64(-z) / a_m) * t));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[a$95$m, 5e-26], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(y / a$95$m), $MachinePrecision] * x + N[(N[((-z) / a$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.00000000000000019e-26

   1. Initial program 92.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 92.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if 5.00000000000000019e-26 < a

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{-z}{a} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{a\_m}, y, \frac{-z}{a\_m} \cdot t\right)\\ t_2 := x \cdot y - t \cdot z\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+256}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (fma (/ x a_m) y (* (/ (- z) a_m) t))) (t_2 (- (* x y) (* t z))))
   (*
    a_s
    (if (<= t_2 (- INFINITY))
      t_1
      (if (<= t_2 1e+256) (/ (fma (- z) t (* x y)) a_m) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = fma((x / a_m), y, ((-z / a_m) * t));
	double t_2 = (x * y) - (t * z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+256) {
		tmp = fma(-z, t, (x * y)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = fma(Float64(x / a_m), y, Float64(Float64(Float64(-z) / a_m) * t))
	t_2 = Float64(Float64(x * y) - Float64(t * z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+256)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x / a$95$m), $MachinePrecision] * y + N[(N[((-z) / a$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+256], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1e256 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 71.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e256

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - t \cdot z \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{-z}{a} \cdot t\right)\\ \mathbf{elif}\;x \cdot y - t \cdot z \leq 10^{+256}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \frac{-z}{a} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{z}{\frac{-a\_m}{t}}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+230}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ z (/ (- a_m) t))))
   (*
    a_s
    (if (<= (* t z) (- INFINITY))
      t_1
      (if (<= (* t z) 1e+230) (/ (fma (- z) t (* x y)) a_m) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = z / (-a_m / t);
	double tmp;
	if ((t * z) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((t * z) <= 1e+230) {
		tmp = fma(-z, t, (x * y)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(z / Float64(Float64(-a_m) / t))
	tmp = 0.0
	if (Float64(t * z) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+230)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(z / N[((-a$95$m) / t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+230], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.0000000000000001e230 < (*.f64 z t)

   1. Initial program 71.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 71.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 95.4

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 97.4%

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

   if -inf.0 < (*.f64 z t) < 1.0000000000000001e230

   1. Initial program 94.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 94.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;t \cdot z \leq 10^{+230}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{z}{\frac{-a\_m}{t}}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 10^{+230}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ z (/ (- a_m) t))))
   (*
    a_s
    (if (<= (* t z) (- INFINITY))
      t_1
      (if (<= (* t z) 1e+230) (/ (- (* x y) (* t z)) a_m) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = z / (-a_m / t);
	double tmp;
	if ((t * z) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((t * z) <= 1e+230) {
		tmp = ((x * y) - (t * z)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = z / (-a_m / t);
	double tmp;
	if ((t * z) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((t * z) <= 1e+230) {
		tmp = ((x * y) - (t * z)) / a_m;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = z / (-a_m / t)
	tmp = 0
	if (t * z) <= -math.inf:
		tmp = t_1
	elif (t * z) <= 1e+230:
		tmp = ((x * y) - (t * z)) / a_m
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(z / Float64(Float64(-a_m) / t))
	tmp = 0.0
	if (Float64(t * z) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(t * z) <= 1e+230)
		tmp = Float64(Float64(Float64(x * y) - Float64(t * z)) / a_m);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = z / (-a_m / t);
	tmp = 0.0;
	if ((t * z) <= -Inf)
		tmp = t_1;
	elseif ((t * z) <= 1e+230)
		tmp = ((x * y) - (t * z)) / a_m;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(z / N[((-a$95$m) / t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 1e+230], N[(N[(N[(x * y), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.0000000000000001e230 < (*.f64 z t)

   1. Initial program 71.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 71.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 95.4

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 97.4%

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

   if -inf.0 < (*.f64 z t) < 1.0000000000000001e230

   1. Initial program 94.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -\infty:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;t \cdot z \leq 10^{+230}:\\ \;\;\;\;\frac{x \cdot y - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{z}{\frac{-a\_m}{t}}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (/ z (/ (- a_m) t))))
   (*
    a_s
    (if (<= (* t z) -5e+33) t_1 (if (<= (* t z) 2e+90) (/ x (/ a_m y)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = z / (-a_m / t);
	double tmp;
	if ((t * z) <= -5e+33) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = x / (a_m / y);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z / (-a_m / t)
    if ((t * z) <= (-5d+33)) then
        tmp = t_1
    else if ((t * z) <= 2d+90) then
        tmp = x / (a_m / y)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = z / (-a_m / t);
	double tmp;
	if ((t * z) <= -5e+33) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = x / (a_m / y);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = z / (-a_m / t)
	tmp = 0
	if (t * z) <= -5e+33:
		tmp = t_1
	elif (t * z) <= 2e+90:
		tmp = x / (a_m / y)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(z / Float64(Float64(-a_m) / t))
	tmp = 0.0
	if (Float64(t * z) <= -5e+33)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+90)
		tmp = Float64(x / Float64(a_m / y));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = z / (-a_m / t);
	tmp = 0.0;
	if ((t * z) <= -5e+33)
		tmp = t_1;
	elseif ((t * z) <= 2e+90)
		tmp = x / (a_m / y);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(z / N[((-a$95$m) / t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], -5e+33], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+90], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e33 or 1.99999999999999993e90 < (*.f64 z t)

   1. Initial program 85.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 85.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 77.9

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

   7. Applied rewrites 77.9%

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

   9. Applied rewrites 79.6%

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

   if -4.99999999999999973e33 < (*.f64 z t) < 1.99999999999999993e90

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 74.1%

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

   9. Applied rewrites 74.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{-t}{a\_m} \cdot z\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{a\_m}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (/ (- t) a_m) z)))
   (*
    a_s
    (if (<= (* t z) -5e+33) t_1 (if (<= (* t z) 2e+90) (/ x (/ a_m y)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-t / a_m) * z;
	double tmp;
	if ((t * z) <= -5e+33) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = x / (a_m / y);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-t / a_m) * z
    if ((t * z) <= (-5d+33)) then
        tmp = t_1
    else if ((t * z) <= 2d+90) then
        tmp = x / (a_m / y)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-t / a_m) * z;
	double tmp;
	if ((t * z) <= -5e+33) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = x / (a_m / y);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (-t / a_m) * z
	tmp = 0
	if (t * z) <= -5e+33:
		tmp = t_1
	elif (t * z) <= 2e+90:
		tmp = x / (a_m / y)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(-t) / a_m) * z)
	tmp = 0.0
	if (Float64(t * z) <= -5e+33)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+90)
		tmp = Float64(x / Float64(a_m / y));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (-t / a_m) * z;
	tmp = 0.0;
	if ((t * z) <= -5e+33)
		tmp = t_1;
	elseif ((t * z) <= 2e+90)
		tmp = x / (a_m / y);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[((-t) / a$95$m), $MachinePrecision] * z), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], -5e+33], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+90], N[(x / N[(a$95$m / y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e33 or 1.99999999999999993e90 < (*.f64 z t)

   1. Initial program 85.5%

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

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -4.99999999999999973e33 < (*.f64 z t) < 1.99999999999999993e90

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 74.1%

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

   9. Applied rewrites 74.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{-t}{a\_m} \cdot z\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (/ (- t) a_m) z)))
   (*
    a_s
    (if (<= (* t z) -5e+33) t_1 (if (<= (* t z) 2e+90) (* (/ y a_m) x) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-t / a_m) * z;
	double tmp;
	if ((t * z) <= -5e+33) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = (y / a_m) * x;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-t / a_m) * z
    if ((t * z) <= (-5d+33)) then
        tmp = t_1
    else if ((t * z) <= 2d+90) then
        tmp = (y / a_m) * x
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-t / a_m) * z;
	double tmp;
	if ((t * z) <= -5e+33) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = (y / a_m) * x;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (-t / a_m) * z
	tmp = 0
	if (t * z) <= -5e+33:
		tmp = t_1
	elif (t * z) <= 2e+90:
		tmp = (y / a_m) * x
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(-t) / a_m) * z)
	tmp = 0.0
	if (Float64(t * z) <= -5e+33)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+90)
		tmp = Float64(Float64(y / a_m) * x);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (-t / a_m) * z;
	tmp = 0.0;
	if ((t * z) <= -5e+33)
		tmp = t_1;
	elseif ((t * z) <= 2e+90)
		tmp = (y / a_m) * x;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[((-t) / a$95$m), $MachinePrecision] * z), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], -5e+33], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+90], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e33 or 1.99999999999999993e90 < (*.f64 z t)

   1. Initial program 85.5%

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

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -4.99999999999999973e33 < (*.f64 z t) < 1.99999999999999993e90

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 74.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \frac{-z}{a\_m} \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{y}{a\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (/ (- z) a_m) t)))
   (*
    a_s
    (if (<= (* t z) -1e+53) t_1 (if (<= (* t z) 2e+90) (* (/ y a_m) x) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-z / a_m) * t;
	double tmp;
	if ((t * z) <= -1e+53) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = (y / a_m) * x;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
real(8) function code(a_s, x, y, z, t, a_m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-z / a_m) * t
    if ((t * z) <= (-1d+53)) then
        tmp = t_1
    else if ((t * z) <= 2d+90) then
        tmp = (y / a_m) * x
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
assert x < y && y < z && z < t && t < a_m;
assert x < y && y < z && z < t && t < a_m;
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (-z / a_m) * t;
	double tmp;
	if ((t * z) <= -1e+53) {
		tmp = t_1;
	} else if ((t * z) <= 2e+90) {
		tmp = (y / a_m) * x;
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
[x, y, z, t, a_m] = sort([x, y, z, t, a_m])
def code(a_s, x, y, z, t, a_m):
	t_1 = (-z / a_m) * t
	tmp = 0
	if (t * z) <= -1e+53:
		tmp = t_1
	elif (t * z) <= 2e+90:
		tmp = (y / a_m) * x
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(-z) / a_m) * t)
	tmp = 0.0
	if (Float64(t * z) <= -1e+53)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+90)
		tmp = Float64(Float64(y / a_m) * x);
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
x, y, z, t, a_m = num2cell(sort([x, y, z, t, a_m])){:}
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (-z / a_m) * t;
	tmp = 0.0;
	if ((t * z) <= -1e+53)
		tmp = t_1;
	elseif ((t * z) <= 2e+90)
		tmp = (y / a_m) * x;
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[((-z) / a$95$m), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t * z), $MachinePrecision], -1e+53], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+90], N[(N[(y / a$95$m), $MachinePrecision] * x), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999999e52 or 1.99999999999999993e90 < (*.f64 z t)

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 85.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -9.9999999999999999e52 < (*.f64 z t) < 1.99999999999999993e90

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 73.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 51.2

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

5. Applied rewrites 51.2%

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

7. Applied rewrites 52.8%

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

8. Final simplification 52.8%

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

Developer Target 1: 91.9% 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 2024294 
(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))