Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Percentage Accurate: 93.6% → 95.1%

Time: 3.2s

Alternatives: 10

Speedup: 1.0×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

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(x + Float64(Float64(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[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

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(x + Float64(Float64(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[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(-y, \frac{z - t}{a \cdot x}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y (- z t)) a))))
   (if (<= t_1 2e+302) t_1 (* (- x) (fma (- y) (/ (- z t) (* a x)) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * (z - t)) / a);
	double tmp;
	if (t_1 <= 2e+302) {
		tmp = t_1;
	} else {
		tmp = -x * fma(-y, ((z - t) / (a * x)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * Float64(z - t)) / a))
	tmp = 0.0
	if (t_1 <= 2e+302)
		tmp = t_1;
	else
		tmp = Float64(Float64(-x) * fma(Float64(-y), Float64(Float64(z - t) / Float64(a * x)), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+302], t$95$1, N[((-x) * N[((-y) * N[(N[(z - t), $MachinePrecision] / N[(a * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 2.0000000000000002e302

   1. Initial program 98.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   if 2.0000000000000002e302 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 85.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift--.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 2: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+112} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -2e+112) (not (<= t_1 4e+58))) (* (/ z a) y) x)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e+112) || !(t_1 <= 4e+58)) {
		tmp = (z / a) * y;
	} else {
		tmp = x;
	}
	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 * (z - t)) / a
    if ((t_1 <= (-2d+112)) .or. (.not. (t_1 <= 4d+58))) then
        tmp = (z / a) * y
    else
        tmp = x
    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 * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e+112) || !(t_1 <= 4e+58)) {
		tmp = (z / a) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if (t_1 <= -2e+112) or not (t_1 <= 4e+58):
		tmp = (z / a) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if ((t_1 <= -2e+112) || !(t_1 <= 4e+58))
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if ((t_1 <= -2e+112) || ~((t_1 <= 4e+58)))
		tmp = (z / a) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+112], N[Not[LessEqual[t$95$1, 4e+58]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.9999999999999999e112 or 3.99999999999999978e58 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 52.4

         \[\leadsto \frac{z}{a} \cdot y \]

   5. Applied rewrites 52.4%

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

   if -1.9999999999999999e112 < (/.f64 (*.f64 y (-.f64 z t)) a) < 3.99999999999999978e58

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 69.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{+112} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 4 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+112}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -2e+112) (/ (* z y) a) (if (<= t_1 4e+58) x (* (/ z a) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -2e+112) {
		tmp = (z * y) / a;
	} else if (t_1 <= 4e+58) {
		tmp = x;
	} else {
		tmp = (z / a) * y;
	}
	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 * (z - t)) / a
    if (t_1 <= (-2d+112)) then
        tmp = (z * y) / a
    else if (t_1 <= 4d+58) then
        tmp = x
    else
        tmp = (z / a) * y
    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 * (z - t)) / a;
	double tmp;
	if (t_1 <= -2e+112) {
		tmp = (z * y) / a;
	} else if (t_1 <= 4e+58) {
		tmp = x;
	} else {
		tmp = (z / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if t_1 <= -2e+112:
		tmp = (z * y) / a
	elif t_1 <= 4e+58:
		tmp = x
	else:
		tmp = (z / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -2e+112)
		tmp = Float64(Float64(z * y) / a);
	elseif (t_1 <= 4e+58)
		tmp = x;
	else
		tmp = Float64(Float64(z / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_1 <= -2e+112)
		tmp = (z * y) / a;
	elseif (t_1 <= 4e+58)
		tmp = x;
	else
		tmp = (z / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+112], N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 4e+58], x, N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.9999999999999999e112

   1. Initial program 94.2%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 51.1

         \[\leadsto \frac{z}{a} \cdot y \]

   5. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \frac{z}{a} \cdot y \]

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

         \[\leadsto \frac{z \cdot y}{a} \]

      7. lower-*.f64 51.2

         \[\leadsto \frac{z \cdot y}{a} \]

   7. Applied rewrites 51.2%

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

   if -1.9999999999999999e112 < (/.f64 (*.f64 y (-.f64 z t)) a) < 3.99999999999999978e58

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{x} \]

   if 3.99999999999999978e58 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 92.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 53.5

         \[\leadsto \frac{z}{a} \cdot y \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{z}{a} \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+72}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e-13)
   (fma (/ y a) z x)
   (if (<= z 6.8e+72) (- x (* y (/ t a))) (fma y (/ z a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e-13) {
		tmp = fma((y / a), z, x);
	} else if (z <= 6.8e+72) {
		tmp = x - (y * (t / a));
	} else {
		tmp = fma(y, (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e-13)
		tmp = fma(Float64(y / a), z, x);
	elseif (z <= 6.8e+72)
		tmp = Float64(x - Float64(y * Float64(t / a)));
	else
		tmp = fma(y, Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e-13], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 6.8e+72], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999999e-13

   1. Initial program 97.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot z + x \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -4.9999999999999999e-13 < z < 6.7999999999999997e72

   1. Initial program 96.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 6.7999999999999997e72 < z

   1. Initial program 94.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 83.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 5: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+33} \lor \neg \left(t \leq 4.2 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.6e+33) (not (<= t 4.2e+200)))
   (* (/ (- y) a) t)
   (fma y (/ z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.6e+33) || !(t <= 4.2e+200)) {
		tmp = (-y / a) * t;
	} else {
		tmp = fma(y, (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.6e+33) || !(t <= 4.2e+200))
		tmp = Float64(Float64(Float64(-y) / a) * t);
	else
		tmp = fma(y, Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.6e+33], N[Not[LessEqual[t, 4.2e+200]], $MachinePrecision]], N[(N[((-y) / a), $MachinePrecision] * t), $MachinePrecision], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.60000000000000021e33 or 4.19999999999999994e200 < t

   1. Initial program 95.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot t}{a} \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if -4.60000000000000021e33 < t < 4.19999999999999994e200

   1. Initial program 96.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+33} \lor \neg \left(t \leq 4.2 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{a}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.6e+33)
   (/ (* (- t) y) a)
   (if (<= t 4.2e+200) (fma y (/ z a) x) (* (/ (- y) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.6e+33) {
		tmp = (-t * y) / a;
	} else if (t <= 4.2e+200) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = (-y / a) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.6e+33)
		tmp = Float64(Float64(Float64(-t) * y) / a);
	elseif (t <= 4.2e+200)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = Float64(Float64(Float64(-y) / a) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.6e+33], N[(N[((-t) * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t, 4.2e+200], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(N[((-y) / a), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.60000000000000021e33

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 66.8

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

   8. Applied rewrites 66.8%

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

   if -4.60000000000000021e33 < t < 4.19999999999999994e200

   1. Initial program 96.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 4.19999999999999994e200 < t

   1. Initial program 92.4%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot t}{a} \]

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{a}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

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(x + Float64(Float64(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[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \frac{z}{a}, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

5. Applied rewrites 69.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 70.8

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

7. Applied rewrites 70.8%

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

Alternative 9: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{a}, z, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

      \[\leadsto \frac{y}{a} \cdot z + x \]

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 70.8

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

5. Applied rewrites 70.8%

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

Alternative 10: 39.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 96.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Applied rewrites 37.2%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (+ x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (+ x (/ (* y (- z t)) a))
       (+ x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / 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 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x + (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / 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 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x + (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x + Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x + (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x + N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t)))))))

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