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

Percentage Accurate: 92.5% → 97.9%

Time: 4.4s

Alternatives: 9

Speedup: 1.1×

• 25.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 97.0

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

4. Applied rewrites 97.0%

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

Alternative 2: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+28} \lor \neg \left(t \leq 7.8 \cdot 10^{-61}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.65e+28) (not (<= t 7.8e-61)))
   (fma y (/ z t) x)
   (/ (* (- z x) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.65e+28) || !(t <= 7.8e-61)) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = ((z - x) * y) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.65e+28) || !(t <= 7.8e-61))
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.65e+28], N[Not[LessEqual[t, 7.8e-61]], $MachinePrecision]], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.65e28 or 7.80000000000000065e-61 < t

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -1.65e28 < t < 7.80000000000000065e-61

   1. Initial program 99.0%

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

   3. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+28} \lor \neg \left(t \leq 7.8 \cdot 10^{-61}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-48} \lor \neg \left(z \leq 1.1 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1e-48) (not (<= z 1.1e-15)))
   (fma y (/ z t) x)
   (* (- 1.0 (/ y t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1e-48) || !(z <= 1.1e-15)) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = (1.0 - (y / t)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1e-48) || !(z <= 1.1e-15))
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1e-48], N[Not[LessEqual[z, 1.1e-15]], $MachinePrecision]], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999997e-49 or 1.09999999999999993e-15 < z

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if -9.9999999999999997e-49 < z < 1.09999999999999993e-15

   1. Initial program 95.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{t}\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-48} \lor \neg \left(z \leq 1.1 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-22}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+105}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.65e-22)
   (* (/ (- z x) t) y)
   (if (<= y 4e+105) (+ x (/ (* y z) t)) (* (/ y t) (- z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.65e-22) {
		tmp = ((z - x) / t) * y;
	} else if (y <= 4e+105) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = (y / t) * (z - 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)
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) :: tmp
    if (y <= (-2.65d-22)) then
        tmp = ((z - x) / t) * y
    else if (y <= 4d+105) then
        tmp = x + ((y * z) / t)
    else
        tmp = (y / t) * (z - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.65e-22) {
		tmp = ((z - x) / t) * y;
	} else if (y <= 4e+105) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = (y / t) * (z - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.65e-22:
		tmp = ((z - x) / t) * y
	elif y <= 4e+105:
		tmp = x + ((y * z) / t)
	else:
		tmp = (y / t) * (z - x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.65e-22)
		tmp = Float64(Float64(Float64(z - x) / t) * y);
	elseif (y <= 4e+105)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = Float64(Float64(y / t) * Float64(z - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.65e-22)
		tmp = ((z - x) / t) * y;
	elseif (y <= 4e+105)
		tmp = x + ((y * z) / t);
	else
		tmp = (y / t) * (z - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.65e-22], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4e+105], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.64999999999999986e-22

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. sub-div N/A

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

      6. *-commutative N/A

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

      7. frac-sub N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 72.1

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 36.5

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 90.0

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

   10. Applied rewrites 90.0%

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

   if -2.64999999999999986e-22 < y < 3.9999999999999998e105

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.6%

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

   if 3.9999999999999998e105 < y

   1. Initial program 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. sub-div N/A

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

      6. *-commutative N/A

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

      7. frac-sub N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-*.f64 58.8

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

   4. Applied rewrites 58.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.0

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

   7. Applied rewrites 38.0%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 86.1

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

   10. Applied rewrites 86.1%

       \[\leadsto \color{blue}{\frac{z - x}{t} \cdot y} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift--.f64 91.0

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

   12. Applied rewrites 91.0%

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

Alternative 5: 69.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-293} \lor \neg \left(z \leq 3.95 \cdot 10^{-81}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5e-293) (not (<= z 3.95e-81)))
   (fma y (/ z t) x)
   (/ (* (- x) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5e-293) || !(z <= 3.95e-81)) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = (-x * y) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5e-293) || !(z <= 3.95e-81))
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = Float64(Float64(Float64(-x) * y) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5e-293], N[Not[LessEqual[z, 3.95e-81]], $MachinePrecision]], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(N[((-x) * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000003e-293 or 3.95000000000000008e-81 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if -5.0000000000000003e-293 < z < 3.95000000000000008e-81

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 63.9

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

   8. Applied rewrites 63.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-293} \lor \neg \left(z \leq 3.95 \cdot 10^{-81}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.6e+30) x (if (<= t 2.6e+62) (/ (* z y) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.6e+30) {
		tmp = x;
	} else if (t <= 2.6e+62) {
		tmp = (z * y) / t;
	} 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)
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) :: tmp
    if (t <= (-5.6d+30)) then
        tmp = x
    else if (t <= 2.6d+62) then
        tmp = (z * y) / t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.6e+30) {
		tmp = x;
	} else if (t <= 2.6e+62) {
		tmp = (z * y) / t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.6e+30:
		tmp = x
	elif t <= 2.6e+62:
		tmp = (z * y) / t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.6e+30)
		tmp = x;
	elseif (t <= 2.6e+62)
		tmp = Float64(Float64(z * y) / t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5.6e+30)
		tmp = x;
	elseif (t <= 2.6e+62)
		tmp = (z * y) / t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.6e+30], x, If[LessEqual[t, 2.6e+62], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -5.59999999999999966e30 or 2.59999999999999984e62 < t

   1. Initial program 85.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 62.3%

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

   if -5.59999999999999966e30 < t < 2.59999999999999984e62

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.5

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

   5. Applied rewrites 50.5%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{-y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.5e+44) (* (/ (- y) t) x) (fma y (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.5e+44) {
		tmp = (-y / t) * x;
	} else {
		tmp = fma(y, (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.5e+44)
		tmp = Float64(Float64(Float64(-y) / t) * x);
	else
		tmp = fma(y, Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.5e+44], N[(N[((-y) / t), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.49999999999999993e44

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -1.49999999999999993e44 < y

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{-y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 69.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 69.9

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

7. Applied rewrites 69.9%

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

Alternative 9: 38.2% accurate, 23.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 34.5%

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

6. Final simplification 34.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 90.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (+ (* x (/ y t)) (* (- z) (/ y t)))))

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