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

Percentage Accurate: 92.9% → 97.7%

Time: 3.6s

Alternatives: 10

Speedup: 1.1×

• 24.6% 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: 92.9% 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: 97.7% 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 92.0%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 98.0

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

4. Applied rewrites 98.0%

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

Alternative 2: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+19} \lor \neg \left(x \leq 3.7 \cdot 10^{+106}\right):\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.6e+19) (not (<= x 3.7e+106)))
   (* (- 1.0 (/ y t)) x)
   (+ x (* (/ y t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e+19) || !(x <= 3.7e+106)) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = x + ((y / t) * z);
	}
	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 ((x <= (-1.6d+19)) .or. (.not. (x <= 3.7d+106))) then
        tmp = (1.0d0 - (y / t)) * x
    else
        tmp = x + ((y / t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e+19) || !(x <= 3.7e+106)) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = x + ((y / t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.6e+19) or not (x <= 3.7e+106):
		tmp = (1.0 - (y / t)) * x
	else:
		tmp = x + ((y / t) * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.6e+19) || !(x <= 3.7e+106))
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	else
		tmp = Float64(x + Float64(Float64(y / t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.6e+19) || ~((x <= 3.7e+106)))
		tmp = (1.0 - (y / t)) * x;
	else
		tmp = x + ((y / t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.6e+19], N[Not[LessEqual[x, 3.7e+106]], $MachinePrecision]], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e19 or 3.69999999999999995e106 < x

   1. Initial program 92.6%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

   if -1.6e19 < x < 3.69999999999999995e106

   1. Initial program 91.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 86.8

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

   8. Applied rewrites 86.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+19} \lor \neg \left(x \leq 3.7 \cdot 10^{+106}\right):\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.18 \cdot 10^{-33} \lor \neg \left(t \leq 11500\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.18e-33) (not (<= t 11500.0)))
   (fma y (/ z t) x)
   (/ (* (- z x) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.18e-33) || !(t <= 11500.0)) {
		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 <= -1.18e-33) || !(t <= 11500.0))
		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, -1.18e-33], N[Not[LessEqual[t, 11500.0]], $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 < -1.18e-33 or 11500 < t

   1. Initial program 86.5%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 82.2%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   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 87.4

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

   7. Applied rewrites 87.4%

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

   if -1.18e-33 < t < 11500

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.18 \cdot 10^{-33} \lor \neg \left(t \leq 11500\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-35} \lor \neg \left(z \leq 2.8 \cdot 10^{-114}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7e-35) (not (<= z 2.8e-114)))
   (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 <= -1.7e-35) || !(z <= 2.8e-114)) {
		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 <= -1.7e-35) || !(z <= 2.8e-114))
		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, -1.7e-35], N[Not[LessEqual[z, 2.8e-114]], $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 < -1.7000000000000001e-35 or 2.8000000000000001e-114 < z

   1. Initial program 89.6%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 81.6%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   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 83.0

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

   7. Applied rewrites 83.0%

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

   if -1.7000000000000001e-35 < z < 2.8000000000000001e-114

   1. Initial program 95.4%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-35} \lor \neg \left(z \leq 2.8 \cdot 10^{-114}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+106}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.6e+19)
   (fma (- x) (/ y t) x)
   (if (<= x 3.7e+106) (+ x (* (/ y t) z)) (* (- 1.0 (/ y t)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e+19) {
		tmp = fma(-x, (y / t), x);
	} else if (x <= 3.7e+106) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = (1.0 - (y / t)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.6e+19)
		tmp = fma(Float64(-x), Float64(y / t), x);
	elseif (x <= 3.7e+106)
		tmp = Float64(x + Float64(Float64(y / t) * z));
	else
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.6e+19], N[((-x) * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 3.7e+106], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e19

   1. Initial program 91.3%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if -1.6e19 < x < 3.69999999999999995e106

   1. Initial program 91.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if 3.69999999999999995e106 < x

   1. Initial program 94.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+106}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.5e-88) (/ (* z y) t) (if (<= z 2.65e-110) x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-88) {
		tmp = (z * y) / t;
	} else if (z <= 2.65e-110) {
		tmp = x;
	} else {
		tmp = y * (z / t);
	}
	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 (z <= (-9.5d-88)) then
        tmp = (z * y) / t
    else if (z <= 2.65d-110) then
        tmp = x
    else
        tmp = y * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.5e-88) {
		tmp = (z * y) / t;
	} else if (z <= 2.65e-110) {
		tmp = x;
	} else {
		tmp = y * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9.5e-88:
		tmp = (z * y) / t
	elif z <= 2.65e-110:
		tmp = x
	else:
		tmp = y * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.5e-88)
		tmp = Float64(Float64(z * y) / t);
	elseif (z <= 2.65e-110)
		tmp = x;
	else
		tmp = Float64(y * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9.5e-88)
		tmp = (z * y) / t;
	elseif (z <= 2.65e-110)
		tmp = x;
	else
		tmp = y * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.5e-88], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.65e-110], x, N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.5e-88

   1. Initial program 92.9%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 56.6

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

   5. Applied rewrites 56.6%

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

   if -9.5e-88 < z < 2.65e-110

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 59.1%

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

   if 2.65e-110 < z

   1. Initial program 87.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 56.8

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

   9. Applied rewrites 56.8%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 26:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.5e-51) x (if (<= t 26.0) (/ (* z y) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-51) {
		tmp = x;
	} else if (t <= 26.0) {
		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 <= (-2.5d-51)) then
        tmp = x
    else if (t <= 26.0d0) 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 <= -2.5e-51) {
		tmp = x;
	} else if (t <= 26.0) {
		tmp = (z * y) / t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.5e-51:
		tmp = x
	elif t <= 26.0:
		tmp = (z * y) / t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.5e-51)
		tmp = x;
	elseif (t <= 26.0)
		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 <= -2.5e-51)
		tmp = x;
	elseif (t <= 26.0)
		tmp = (z * y) / t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.5e-51], x, If[LessEqual[t, 26.0], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.50000000000000002e-51 or 26 < t

   1. Initial program 86.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.2%

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

   if -2.50000000000000002e-51 < t < 26

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 53.1

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

   5. Applied rewrites 53.1%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 26:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{-y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.2e+232) (* (/ (- y) t) x) (fma y (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e+232) {
		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 <= -5.2e+232)
		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, -5.2e+232], 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 < -5.19999999999999947e232

   1. Initial program 76.4%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   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 83.5

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

   8. Applied rewrites 83.5%

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

   if -5.19999999999999947e232 < y

   1. Initial program 92.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 75.3%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   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 74.7

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

   7. Applied rewrites 74.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{-y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% 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 92.0%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 73.4%

   \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
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 73.3

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

7. Applied rewrites 73.3%

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

Alternative 10: 39.0% 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 92.0%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 39.9%

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

6. Final simplification 39.9%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 91.0% 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 2025051 
(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)))