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

Percentage Accurate: 92.4% → 97.5%

Time: 7.0s

Alternatives: 11

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: 92.4% 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: 97.5% 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 92.4%

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

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

   4. lift--.f64 N/A

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

   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 97.5

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

3. Applied rewrites 97.5%

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

Alternative 2: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+125}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -45.0)
   (* (/ (- z x) t) y)
   (if (<= y 1.3e+125) (+ x (/ (* y z) t)) (* (/ y t) (- z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -45.0) {
		tmp = ((z - x) / t) * y;
	} else if (y <= 1.3e+125) {
		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 <= (-45.0d0)) then
        tmp = ((z - x) / t) * y
    else if (y <= 1.3d+125) 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 <= -45.0) {
		tmp = ((z - x) / t) * y;
	} else if (y <= 1.3e+125) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = (y / t) * (z - x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -45.0:
		tmp = ((z - x) / t) * y
	elif y <= 1.3e+125:
		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 <= -45.0)
		tmp = Float64(Float64(Float64(z - x) / t) * y);
	elseif (y <= 1.3e+125)
		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 <= -45.0)
		tmp = ((z - x) / t) * y;
	elseif (y <= 1.3e+125)
		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, -45.0], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.3e+125], 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 < -45

   1. Initial program 86.0%

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

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

      4. lift--.f64 N/A

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

      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 96.6

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 74.8

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

   6. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 82.7

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

   8. Applied rewrites 82.7%

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

   if -45 < y < 1.30000000000000002e125

   1. Initial program 97.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 83.6%

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

   if 1.30000000000000002e125 < y

   1. Initial program 82.5%

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

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

      4. lift--.f64 N/A

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

      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 95.6

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

   3. Applied rewrites 95.6%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 77.7

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

   6. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 86.9

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

   8. Applied rewrites 86.9%

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

Alternative 3: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+140}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{t - y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= z -1.15e+140)
     (* (/ y t) (- z x))
     (if (<= z -1.15e-137) t_1 (if (<= z 1.3e-59) (* (/ (- t y) t) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (z <= -1.15e+140) {
		tmp = (y / t) * (z - x);
	} else if (z <= -1.15e-137) {
		tmp = t_1;
	} else if (z <= 1.3e-59) {
		tmp = ((t - y) / t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (z <= -1.15e+140)
		tmp = Float64(Float64(y / t) * Float64(z - x));
	elseif (z <= -1.15e-137)
		tmp = t_1;
	elseif (z <= 1.3e-59)
		tmp = Float64(Float64(Float64(t - y) / t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.15e+140], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.15e-137], t$95$1, If[LessEqual[z, 1.3e-59], N[(N[(N[(t - y), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999995e140

   1. Initial program 88.2%

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

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

      4. lift--.f64 N/A

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

      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 97.4

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

   3. Applied rewrites 97.4%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 66.8

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

   6. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if -1.14999999999999995e140 < z < -1.15000000000000004e-137 or 1.29999999999999999e-59 < z

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 79.6%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   5. 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 80.0

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

   6. Applied rewrites 80.0%

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

   if -1.15000000000000004e-137 < z < 1.29999999999999999e-59

   1. Initial program 94.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   3. 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 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 4: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -200:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -200.0)
   (* (/ (- z x) t) y)
   (if (<= y 1.1e+145) (fma y (/ z t) x) (* (/ y t) (- z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -200.0) {
		tmp = ((z - x) / t) * y;
	} else if (y <= 1.1e+145) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = (y / t) * (z - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -200.0)
		tmp = Float64(Float64(Float64(z - x) / t) * y);
	elseif (y <= 1.1e+145)
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = Float64(Float64(y / t) * Float64(z - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -200.0], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.1e+145], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -200

   1. Initial program 85.9%

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

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

      4. lift--.f64 N/A

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

      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 96.6

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 74.7

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

   6. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 82.7

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

   8. Applied rewrites 82.7%

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

   if -200 < y < 1.10000000000000004e145

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 83.0%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   5. 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 78.9

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

   6. Applied rewrites 78.9%

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

   if 1.10000000000000004e145 < y

   1. Initial program 81.6%

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

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

      4. lift--.f64 N/A

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

      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 95.4

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

   3. Applied rewrites 95.4%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 77.6

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

   6. Applied rewrites 77.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 87.6

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

   8. Applied rewrites 87.6%

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

Alternative 5: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{t - y}{t} \cdot x\\ \mathbf{elif}\;t \leq 1750:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.5e+80)
   (* (/ (- t y) t) x)
   (if (<= t 1750.0) (/ (* (- z x) y) t) (fma y (/ z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.5e+80) {
		tmp = ((t - y) / t) * x;
	} else if (t <= 1750.0) {
		tmp = ((z - x) * y) / t;
	} else {
		tmp = fma(y, (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.5e+80)
		tmp = Float64(Float64(Float64(t - y) / t) * x);
	elseif (t <= 1750.0)
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	else
		tmp = fma(y, Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.5e+80], N[(N[(N[(t - y), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1750.0], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.49999999999999967e80

   1. Initial program 82.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   3. 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 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if -5.49999999999999967e80 < t < 1750

   1. Initial program 98.2%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   3. 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 80.0

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

   4. Applied rewrites 80.0%

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

   if 1750 < t

   1. Initial program 87.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 81.0%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   5. 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 87.1

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

   6. Applied rewrites 87.1%

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

Alternative 6: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{t - y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= z -1.15e-137) t_1 (if (<= z 1.3e-59) (* (/ (- t y) t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (z <= -1.15e-137) {
		tmp = t_1;
	} else if (z <= 1.3e-59) {
		tmp = ((t - y) / t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (z <= -1.15e-137)
		tmp = t_1;
	elseif (z <= 1.3e-59)
		tmp = Float64(Float64(Float64(t - y) / t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.15e-137], t$95$1, If[LessEqual[z, 1.3e-59], N[(N[(N[(t - y), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000004e-137 or 1.29999999999999999e-59 < z

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 80.7%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   5. 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 81.1

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

   6. Applied rewrites 81.1%

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

   if -1.15000000000000004e-137 < z < 1.29999999999999999e-59

   1. Initial program 94.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y}{t}\right)} \]
   3. 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 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 7: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{-262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-200}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= z -2.15e-262) t_1 (if (<= z 1.3e-200) (/ (* (- x) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (z <= -2.15e-262) {
		tmp = t_1;
	} else if (z <= 1.3e-200) {
		tmp = (-x * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (z <= -2.15e-262)
		tmp = t_1;
	elseif (z <= 1.3e-200)
		tmp = Float64(Float64(Float64(-x) * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.15e-262], t$95$1, If[LessEqual[z, 1.3e-200], N[(N[((-x) * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1500000000000001e-262 or 1.29999999999999995e-200 < z

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 76.1%

      \[\leadsto x + \frac{y \cdot \color{blue}{z}}{t} \]
   5. 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

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

   6. Applied rewrites 76.3%

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

   if -2.1500000000000001e-262 < z < 1.29999999999999995e-200

   1. Initial program 92.7%

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

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

      4. lift--.f64 N/A

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

      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 95.3

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

   3. Applied rewrites 95.3%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 46.9

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

   6. Applied rewrites 46.9%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.8

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

   9. Applied rewrites 42.8%

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

Alternative 8: 52.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-262}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)))
   (if (<= z -6.5e-20)
     t_1
     (if (<= z -2.15e-262)
       x
       (if (<= z 2.7e-130) (/ (* (- x) y) t) (if (<= z 2.1e-33) x t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (z <= -6.5e-20) {
		tmp = t_1;
	} else if (z <= -2.15e-262) {
		tmp = x;
	} else if (z <= 2.7e-130) {
		tmp = (-x * y) / t;
	} else if (z <= 2.1e-33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_1
    real(8) :: tmp
    t_1 = (z * y) / t
    if (z <= (-6.5d-20)) then
        tmp = t_1
    else if (z <= (-2.15d-262)) then
        tmp = x
    else if (z <= 2.7d-130) then
        tmp = (-x * y) / t
    else if (z <= 2.1d-33) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (z <= -6.5e-20) {
		tmp = t_1;
	} else if (z <= -2.15e-262) {
		tmp = x;
	} else if (z <= 2.7e-130) {
		tmp = (-x * y) / t;
	} else if (z <= 2.1e-33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) / t
	tmp = 0
	if z <= -6.5e-20:
		tmp = t_1
	elif z <= -2.15e-262:
		tmp = x
	elif z <= 2.7e-130:
		tmp = (-x * y) / t
	elif z <= 2.1e-33:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / t)
	tmp = 0.0
	if (z <= -6.5e-20)
		tmp = t_1;
	elseif (z <= -2.15e-262)
		tmp = x;
	elseif (z <= 2.7e-130)
		tmp = Float64(Float64(Float64(-x) * y) / t);
	elseif (z <= 2.1e-33)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) / t;
	tmp = 0.0;
	if (z <= -6.5e-20)
		tmp = t_1;
	elseif (z <= -2.15e-262)
		tmp = x;
	elseif (z <= 2.7e-130)
		tmp = (-x * y) / t;
	elseif (z <= 2.1e-33)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -6.5e-20], t$95$1, If[LessEqual[z, -2.15e-262], x, If[LessEqual[z, 2.7e-130], N[(N[((-x) * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.1e-33], x, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.50000000000000032e-20 or 2.1e-33 < z

   1. Initial program 90.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   3. 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 55.1

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

   4. Applied rewrites 55.1%

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

   if -6.50000000000000032e-20 < z < -2.1500000000000001e-262 or 2.69999999999999991e-130 < z < 2.1e-33

   1. Initial program 95.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 48.6%

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

   if -2.1500000000000001e-262 < z < 2.69999999999999991e-130

   1. Initial program 93.6%

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

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

      4. lift--.f64 N/A

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

      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 95.6

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

   3. Applied rewrites 95.6%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 48.2

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

   6. Applied rewrites 48.2%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.5

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

   9. Applied rewrites 41.5%

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

Alternative 9: 50.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-262}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-128}:\\ \;\;\;\;-x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)))
   (if (<= z -6.5e-20)
     t_1
     (if (<= z -2.15e-262)
       x
       (if (<= z 3.2e-128) (- (* x (/ y t))) (if (<= z 2.1e-33) x t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (z <= -6.5e-20) {
		tmp = t_1;
	} else if (z <= -2.15e-262) {
		tmp = x;
	} else if (z <= 3.2e-128) {
		tmp = -(x * (y / t));
	} else if (z <= 2.1e-33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_1
    real(8) :: tmp
    t_1 = (z * y) / t
    if (z <= (-6.5d-20)) then
        tmp = t_1
    else if (z <= (-2.15d-262)) then
        tmp = x
    else if (z <= 3.2d-128) then
        tmp = -(x * (y / t))
    else if (z <= 2.1d-33) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (z <= -6.5e-20) {
		tmp = t_1;
	} else if (z <= -2.15e-262) {
		tmp = x;
	} else if (z <= 3.2e-128) {
		tmp = -(x * (y / t));
	} else if (z <= 2.1e-33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) / t
	tmp = 0
	if z <= -6.5e-20:
		tmp = t_1
	elif z <= -2.15e-262:
		tmp = x
	elif z <= 3.2e-128:
		tmp = -(x * (y / t))
	elif z <= 2.1e-33:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / t)
	tmp = 0.0
	if (z <= -6.5e-20)
		tmp = t_1;
	elseif (z <= -2.15e-262)
		tmp = x;
	elseif (z <= 3.2e-128)
		tmp = Float64(-Float64(x * Float64(y / t)));
	elseif (z <= 2.1e-33)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) / t;
	tmp = 0.0;
	if (z <= -6.5e-20)
		tmp = t_1;
	elseif (z <= -2.15e-262)
		tmp = x;
	elseif (z <= 3.2e-128)
		tmp = -(x * (y / t));
	elseif (z <= 2.1e-33)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -6.5e-20], t$95$1, If[LessEqual[z, -2.15e-262], x, If[LessEqual[z, 3.2e-128], (-N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision]), If[LessEqual[z, 2.1e-33], x, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.50000000000000032e-20 or 2.1e-33 < z

   1. Initial program 90.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   3. 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 55.1

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

   4. Applied rewrites 55.1%

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

   if -6.50000000000000032e-20 < z < -2.1500000000000001e-262 or 3.1999999999999998e-128 < z < 2.1e-33

   1. Initial program 95.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 48.5%

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

   if -2.1500000000000001e-262 < z < 3.1999999999999998e-128

   1. Initial program 93.6%

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

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

      4. lift--.f64 N/A

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

      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 95.7

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]
   5. 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 48.0

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

   6. Applied rewrites 48.0%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.6

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

   9. Applied rewrites 41.6%

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

Alternative 10: 50.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)))
   (if (<= z -6.5e-20) t_1 (if (<= z 2.1e-33) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (z <= -6.5e-20) {
		tmp = t_1;
	} else if (z <= 2.1e-33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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) :: t_1
    real(8) :: tmp
    t_1 = (z * y) / t
    if (z <= (-6.5d-20)) then
        tmp = t_1
    else if (z <= 2.1d-33) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (z <= -6.5e-20) {
		tmp = t_1;
	} else if (z <= 2.1e-33) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) / t
	tmp = 0
	if z <= -6.5e-20:
		tmp = t_1
	elif z <= 2.1e-33:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / t)
	tmp = 0.0
	if (z <= -6.5e-20)
		tmp = t_1;
	elseif (z <= 2.1e-33)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) / t;
	tmp = 0.0;
	if (z <= -6.5e-20)
		tmp = t_1;
	elseif (z <= 2.1e-33)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -6.5e-20], t$95$1, If[LessEqual[z, 2.1e-33], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000032e-20 or 2.1e-33 < z

   1. Initial program 90.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   3. 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 55.1

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

   4. Applied rewrites 55.1%

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

   if -6.50000000000000032e-20 < z < 2.1e-33

   1. Initial program 94.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 49.1%

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

Alternative 11: 38.6% accurate, 12.7× 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 92.4%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 38.6%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64
  (+ x (/ (* y (- z x)) t)))