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

Percentage Accurate: 91.4% → 94.4%

Time: 3.9s

Alternatives: 5

Speedup: N/A×

• 28.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 91.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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: 94.4% accurate, N/A× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := \left(-z\right) \cdot \frac{t}{a\_m}\\ t_2 := \frac{x \cdot y - z \cdot t}{a\_m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a\_m}, x, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+178}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (- z) (/ t a_m))) (t_2 (/ (- (* x y) (* z t)) a_m)))
   (*
    a_s
    (if (<= t_2 (- INFINITY))
      (fma (/ y a_m) x t_1)
      (if (<= t_2 2e+178) t_2 (fma (/ 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 * (t / a_m);
	double t_2 = ((x * y) - (z * t)) / a_m;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((y / a_m), x, t_1);
	} else if (t_2 <= 2e+178) {
		tmp = t_2;
	} else {
		tmp = fma((x / a_m), y, 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(-z) * Float64(t / a_m))
	t_2 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a_m)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(y / a_m), x, t_1);
	elseif (t_2 <= 2e+178)
		tmp = t_2;
	else
		tmp = fma(Float64(x / a_m), y, 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[(t / a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / a$95$m), $MachinePrecision] * x + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+178], t$95$2, N[(N[(x / a$95$m), $MachinePrecision] * y + t$95$1), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -inf.0

   1. Initial program 82.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. div-add N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

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

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*r* N/A

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

      23. *-commutative N/A

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

   4. Applied rewrites 86.7%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 2.0000000000000001e178

   1. Initial program 99.0%

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

   if 2.0000000000000001e178 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. div-add N/A

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

      11. associate-*l/ N/A

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

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

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. associate-*r/ N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 94.0%

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

Alternative 2: 95.8% accurate, N/A× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+305} \lor \neg \left(t\_1 \leq 10^{+93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a\_m}, y, \left(-z\right) \cdot \frac{t}{a\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (*
    a_s
    (if (or (<= t_1 -4e+305) (not (<= t_1 1e+93)))
      (fma (/ x a_m) y (* (- z) (/ t a_m)))
      (/ t_1 a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -4e+305) || !(t_1 <= 1e+93)) {
		tmp = fma((x / a_m), y, (-z * (t / a_m)));
	} else {
		tmp = t_1 / a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -4e+305) || !(t_1 <= 1e+93))
		tmp = fma(Float64(x / a_m), y, Float64(Float64(-z) * Float64(t / a_m)));
	else
		tmp = Float64(t_1 / a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[Or[LessEqual[t$95$1, -4e+305], N[Not[LessEqual[t$95$1, 1e+93]], $MachinePrecision]], N[(N[(x / a$95$m), $MachinePrecision] * y + N[((-z) * N[(t / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -3.9999999999999998e305 or 1.00000000000000004e93 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 82.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. *-commutative N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. div-add N/A

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

      11. associate-*l/ N/A

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

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

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. associate-*r/ N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 95.7%

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

   if -3.9999999999999998e305 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000004e93

   1. Initial program 99.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -4 \cdot 10^{+305} \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, \left(-z\right) \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, N/A× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\\\ [x, y, z, t, a_m] = \mathsf{sort}([x, y, z, t, a_m])\\ \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-z\right) \cdot t\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{a\_m} \cdot t, z, \frac{x \cdot y}{a\_m}\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* z t) 5e+270)
    (/ (fma y x (* (- z) t)) a_m)
    (fma (* (/ -1.0 a_m) t) z (/ (* x y) a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
assert(x < y && y < z && z < t && t < a_m);
assert(x < y && y < z && z < t && t < a_m);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((z * t) <= 5e+270) {
		tmp = fma(y, x, (-z * t)) / a_m;
	} else {
		tmp = fma(((-1.0 / a_m) * t), z, ((x * y) / a_m));
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
x, y, z, t, a_m = sort([x, y, z, t, a_m])
x, y, z, t, a_m = sort([x, y, z, t, a_m])
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(z * t) <= 5e+270)
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a_m);
	else
		tmp = fma(Float64(Float64(-1.0 / a_m) * t), z, Float64(Float64(x * y) / a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a_m should be sorted in increasing order before calling this function.
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(z * t), $MachinePrecision], 5e+270], N[(N[(y * x + N[((-z) * t), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(N[(-1.0 / a$95$m), $MachinePrecision] * t), $MachinePrecision] * z + N[(N[(x * y), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.99999999999999976e270

   1. Initial program 96.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if 4.99999999999999976e270 < (*.f64 z t)

   1. Initial program 63.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. frac-2neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. div-add-rev N/A

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

      13. associate-*r/ N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 N/A

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

   4. Applied rewrites 92.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

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

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 92.4

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 92.4

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

   6. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{a} \cdot t, z, \frac{x \cdot y}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

   \[\frac{x \cdot y - z \cdot t}{a} \]
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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 93.0

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

4. Applied rewrites 93.0%

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

Alternative 5: 91.4% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

a\_m =     private
a\_s =     private
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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a_m
    code = a_s * (((x * y) - (z * t)) / a_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

Reproduce

herbie shell --seed 2025057 
(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))