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

Percentage Accurate: 91.4% → 96.1%

Time: 5.9s

Alternatives: 6

Speedup: 0.5×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;\left(\frac{\frac{x}{z} - \frac{t}{y}}{a} \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+306)))
     (* (* (/ (- (/ x z) (/ t y)) a) z) y)
     (/ t_1 a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+306)) {
		tmp = ((((x / z) - (t / y)) / a) * z) * y;
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+306)) {
		tmp = ((((x / z) - (t / y)) / a) * z) * y;
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+306):
		tmp = ((((x / z) - (t / y)) / a) * z) * y
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+306))
		tmp = Float64(Float64(Float64(Float64(Float64(x / z) - Float64(t / y)) / a) * z) * y);
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+306)))
		tmp = ((((x / z) - (t / y)) / a) * z) * y;
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+306]], $MachinePrecision]], N[(N[(N[(N[(N[(x / z), $MachinePrecision] - N[(t / y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2.00000000000000003e306 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 55.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   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. add-sqr-sqrt N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 28.2

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

   4. Applied rewrites 28.2%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in z around inf

      \[\leadsto \left(z \cdot \left(-1 \cdot \frac{t}{a \cdot y} + \frac{x}{a \cdot z}\right)\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 82.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000003e306

   1. Initial program 98.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;\left(\frac{\frac{x}{z} - \frac{t}{y}}{a} \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000000:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) -5000000000.0)
   (* (/ (- z) a) t)
   (if (<= (* z t) 4e-35)
     (* (/ x a) y)
     (if (<= (* z t) 5e+273) (/ (* (- z) t) a) (* (- z) (/ t a))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -5000000000.0) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= 4e-35) {
		tmp = (x / a) * y;
	} else if ((z * t) <= 5e+273) {
		tmp = (-z * t) / a;
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if ((z * t) <= (-5000000000.0d0)) then
        tmp = (-z / a) * t
    else if ((z * t) <= 4d-35) then
        tmp = (x / a) * y
    else if ((z * t) <= 5d+273) then
        tmp = (-z * t) / a
    else
        tmp = -z * (t / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -5000000000.0) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= 4e-35) {
		tmp = (x / a) * y;
	} else if ((z * t) <= 5e+273) {
		tmp = (-z * t) / a;
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -5000000000.0:
		tmp = (-z / a) * t
	elif (z * t) <= 4e-35:
		tmp = (x / a) * y
	elif (z * t) <= 5e+273:
		tmp = (-z * t) / a
	else:
		tmp = -z * (t / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= -5000000000.0)
		tmp = Float64(Float64(Float64(-z) / a) * t);
	elseif (Float64(z * t) <= 4e-35)
		tmp = Float64(Float64(x / a) * y);
	elseif (Float64(z * t) <= 5e+273)
		tmp = Float64(Float64(Float64(-z) * t) / a);
	else
		tmp = Float64(Float64(-z) * Float64(t / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -5000000000.0)
		tmp = (-z / a) * t;
	elseif ((z * t) <= 4e-35)
		tmp = (x / a) * y;
	elseif ((z * t) <= 5e+273)
		tmp = (-z * t) / a;
	else
		tmp = -z * (t / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], -5000000000.0], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e-35], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+273], N[(N[((-z) * t), $MachinePrecision] / a), $MachinePrecision], N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -5e9

   1. Initial program 79.8%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 45.4

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 86.4%

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

   if -5e9 < (*.f64 z t) < 4.00000000000000003e-35

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 79.2%

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

   if 4.00000000000000003e-35 < (*.f64 z t) < 4.99999999999999961e273

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 81.8%

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

   if 4.99999999999999961e273 < (*.f64 z t)

   1. Initial program 55.1%

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

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\left(-z\right) \cdot \frac{t}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000000:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{\left(-z\right) \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* (/ (- z) a) t)
   (if (<= (* z t) 5e+273) (/ (- (* x y) (* z t)) a) (* (- z) (/ t a)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= 5e+273) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-z / a) * t;
	} else if ((z * t) <= 5e+273) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -z * (t / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-z / a) * t
	elif (z * t) <= 5e+273:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -z * (t / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-z) / a) * t);
	elseif (Float64(z * t) <= 5e+273)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(-z) * Float64(t / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-z / a) * t;
	elseif ((z * t) <= 5e+273)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -z * (t / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+273], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.4%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 42.6

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 z t) < 4.99999999999999961e273

   1. Initial program 95.0%

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

   if 4.99999999999999961e273 < (*.f64 z t)

   1. Initial program 55.1%

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

   5. Applied rewrites 95.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000000 \lor \neg \left(z \cdot t \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* z t) -5000000000.0) (not (<= (* z t) 4e-35)))
   (* (/ (- z) a) t)
   (* (/ x a) y)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5000000000.0) || !((z * t) <= 4e-35)) {
		tmp = (-z / a) * t;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if (((z * t) <= (-5000000000.0d0)) .or. (.not. ((z * t) <= 4d-35))) then
        tmp = (-z / a) * t
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5000000000.0) || !((z * t) <= 4e-35)) {
		tmp = (-z / a) * t;
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -5000000000.0) or not ((z * t) <= 4e-35):
		tmp = (-z / a) * t
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -5000000000.0) || !(Float64(z * t) <= 4e-35))
		tmp = Float64(Float64(Float64(-z) / a) * t);
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z * t) <= -5000000000.0) || ~(((z * t) <= 4e-35)))
		tmp = (-z / a) * t;
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5000000000.0], N[Not[LessEqual[N[(z * t), $MachinePrecision], 4e-35]], $MachinePrecision]], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e9 or 4.00000000000000003e-35 < (*.f64 z t)

   1. Initial program 85.4%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 42.1

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 82.8%

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

   if -5e9 < (*.f64 z t) < 4.00000000000000003e-35

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 79.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000000 \lor \neg \left(z \cdot t \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000000 \lor \neg \left(z \cdot t \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (* z t) -5000000000.0) (not (<= (* z t) 4e-35)))
   (* (- z) (/ t a))
   (* (/ x a) y)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5000000000.0) || !((z * t) <= 4e-35)) {
		tmp = -z * (t / a);
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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(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
    real(8) :: tmp
    if (((z * t) <= (-5000000000.0d0)) .or. (.not. ((z * t) <= 4d-35))) then
        tmp = -z * (t / a)
    else
        tmp = (x / a) * y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * t) <= -5000000000.0) || !((z * t) <= 4e-35)) {
		tmp = -z * (t / a);
	} else {
		tmp = (x / a) * y;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if ((z * t) <= -5000000000.0) or not ((z * t) <= 4e-35):
		tmp = -z * (t / a)
	else:
		tmp = (x / a) * y
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(z * t) <= -5000000000.0) || !(Float64(z * t) <= 4e-35))
		tmp = Float64(Float64(-z) * Float64(t / a));
	else
		tmp = Float64(Float64(x / a) * y);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z * t) <= -5000000000.0) || ~(((z * t) <= 4e-35)))
		tmp = -z * (t / a);
	else
		tmp = (x / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5000000000.0], N[Not[LessEqual[N[(z * t), $MachinePrecision], 4e-35]], $MachinePrecision]], N[((-z) * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e9 or 4.00000000000000003e-35 < (*.f64 z t)

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.0%

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

   if -5e9 < (*.f64 z t) < 4.00000000000000003e-35

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 79.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5000000000 \lor \neg \left(z \cdot t \leq 4 \cdot 10^{-35}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{x}{a} \cdot y \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* (/ x a) y))

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a 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(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 / a) * y
end function

Java

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

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (x / a) * y

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(x / a) * y)
end

MATLAB

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

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(x / a), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 88.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 47.4%

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

Developer Target 1: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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
    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 2025021 
(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))