Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.8% → 99.9%

Time: 8.5s

Alternatives: 13

Speedup: 1.2×

• 26.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

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) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

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) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z + y, t\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma y 5.0 (* (fma 2.0 (+ z y) t) x)))

C

double code(double x, double y, double z, double t) {
	return fma(y, 5.0, (fma(2.0, (z + y), t) * x));
}

Julia

function code(x, y, z, t)
	return fma(y, 5.0, Float64(fma(2.0, Float64(z + y), t) * x))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

   4. lower-fma.f64 100.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   7. lower-*.f64 100.0

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

   8. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

   9. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

   11. associate-+l+ N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

   14. count-2 N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x\right) \]

   15. lower-fma.f64 100.0

       \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x\right) \]

   16. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \cdot x\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

   18. lower-+.f64 100.0

       \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z + y, t\right) \cdot x\right)} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+15} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e+15) (not (<= x 2.5)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* (fma 2.0 z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e+15) || !(x <= 2.5)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, (fma(2.0, z, t) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.5e+15) || !(x <= 2.5))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(fma(2.0, z, t) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.5e+15], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e15 or 2.5 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 3.2

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 99.8

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -2.5e15 < x < 2.5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x\right) \]

      15. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \cdot x\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

      18. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z + y, t\right) \cdot x\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(t + 2 \cdot z\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z + t\right)} \cdot x\right) \]

      2. lower-fma.f64 98.6

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   7. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+15} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-27} \lor \neg \left(x \leq 9400\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.2e-27) (not (<= x 9400.0)))
   (* (fma 2.0 (+ z y) t) x)
   (fma (fma 2.0 y t) x (* 5.0 y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.2e-27) || !(x <= 9400.0)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(fma(2.0, y, t), x, (5.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.2e-27) || !(x <= 9400.0))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(fma(2.0, y, t), x, Float64(5.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.2e-27], N[Not[LessEqual[x, 9400.0]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * y + t), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999987e-27 or 9400 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 3.5

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 99.1

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -2.19999999999999987e-27 < x < 9400

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot y\right) + 5 \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot y\right) \cdot x} + 5 \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + 2 \cdot y, x, 5 \cdot y\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot y + t}, x, 5 \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, y, t\right)}, x, 5 \cdot y\right) \]

      6. lower-*.f64 82.5

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, \color{blue}{5 \cdot y}\right) \]

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-27} \lor \neg \left(x \leq 9400\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -3.35 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-59}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -3.35e+31)
     t_1
     (if (<= y -4.2e-59)
       (* (* z x) 2.0)
       (if (<= y 1.15e+14) (* (fma 2.0 y t) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -3.35e+31) {
		tmp = t_1;
	} else if (y <= -4.2e-59) {
		tmp = (z * x) * 2.0;
	} else if (y <= 1.15e+14) {
		tmp = fma(2.0, y, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -3.35e+31)
		tmp = t_1;
	elseif (y <= -4.2e-59)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (y <= 1.15e+14)
		tmp = Float64(fma(2.0, y, t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -3.35e+31], t$95$1, If[LessEqual[y, -4.2e-59], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 1.15e+14], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.35000000000000008e31 or 1.15e14 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + 2 \cdot x\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + 2 \cdot x\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot x + 5\right)} \cdot y \]

      4. lower-fma.f64 79.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -3.35000000000000008e31 < y < -4.19999999999999993e-59

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

      4. lower-*.f64 60.7

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]

   if -4.19999999999999993e-59 < y < 1.15e14

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 12.3

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 12.3%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 90.8

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(t + 2 \cdot y\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 55.8%

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

Alternative 5: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48} \lor \neg \left(x \leq 2.7 \cdot 10^{-125}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.25e-48) (not (<= x 2.7e-125)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* (* 2.0 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e-48) || !(x <= 2.7e-125)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, ((2.0 * z) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.25e-48) || !(x <= 2.7e-125))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(Float64(2.0 * z) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.25e-48], N[Not[LessEqual[x, 2.7e-125]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(N[(2.0 * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.24999999999999994e-48 or 2.6999999999999998e-125 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 6.4

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 6.4%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 95.6

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -2.24999999999999994e-48 < x < 2.6999999999999998e-125

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x\right) \]

      15. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \cdot x\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

      18. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z + y, t\right) \cdot x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 78.6

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]

   7. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48} \lor \neg \left(x \leq 2.7 \cdot 10^{-125}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-42}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-108}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.25e-42)
   (* (* z x) 2.0)
   (if (<= x 6e-108) (* 5.0 y) (if (<= x 3.1e+68) (* t x) (* (* x y) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.25e-42) {
		tmp = (z * x) * 2.0;
	} else if (x <= 6e-108) {
		tmp = 5.0 * y;
	} else if (x <= 3.1e+68) {
		tmp = t * x;
	} else {
		tmp = (x * y) * 2.0;
	}
	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 <= (-1.25d-42)) then
        tmp = (z * x) * 2.0d0
    else if (x <= 6d-108) then
        tmp = 5.0d0 * y
    else if (x <= 3.1d+68) then
        tmp = t * x
    else
        tmp = (x * y) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.25e-42) {
		tmp = (z * x) * 2.0;
	} else if (x <= 6e-108) {
		tmp = 5.0 * y;
	} else if (x <= 3.1e+68) {
		tmp = t * x;
	} else {
		tmp = (x * y) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.25e-42:
		tmp = (z * x) * 2.0
	elif x <= 6e-108:
		tmp = 5.0 * y
	elif x <= 3.1e+68:
		tmp = t * x
	else:
		tmp = (x * y) * 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.25e-42)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (x <= 6e-108)
		tmp = Float64(5.0 * y);
	elseif (x <= 3.1e+68)
		tmp = Float64(t * x);
	else
		tmp = Float64(Float64(x * y) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.25e-42)
		tmp = (z * x) * 2.0;
	elseif (x <= 6e-108)
		tmp = 5.0 * y;
	elseif (x <= 3.1e+68)
		tmp = t * x;
	else
		tmp = (x * y) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.25e-42], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, 6e-108], N[(5.0 * y), $MachinePrecision], If[LessEqual[x, 3.1e+68], N[(t * x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25000000000000001e-42

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

      4. lower-*.f64 42.2

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

   5. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]

   if -1.25000000000000001e-42 < x < 5.99999999999999986e-108

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 60.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if 5.99999999999999986e-108 < x < 3.0999999999999998e68

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

   if 3.0999999999999998e68 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(2 \cdot y + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + 2 \cdot z\right) + 5 \cdot y} \]

      2. distribute-lft-out N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(y + z\right)\right)} + 5 \cdot y \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot \left(y + z\right)} + 5 \cdot y \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot x\right)} \cdot \left(y + z\right) + 5 \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, y + z, 5 \cdot y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot x}, y + z, 5 \cdot y\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot x, \color{blue}{z + y}, 5 \cdot y\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot x, \color{blue}{z + y}, 5 \cdot y\right) \]

      9. lower-*.f64 80.1

         \[\leadsto \mathsf{fma}\left(2 \cdot x, z + y, \color{blue}{5 \cdot y}\right) \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, z + y, 5 \cdot y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot \left(y + z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto \left(\left(z + y\right) \cdot x\right) \cdot \color{blue}{2} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(x \cdot y\right) \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 54.9%

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

Alternative 7: 46.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-108}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.25e-48)
   (* t x)
   (if (<= x 6e-108) (* 5.0 y) (if (<= x 3.1e+68) (* t x) (* (* x y) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.25e-48) {
		tmp = t * x;
	} else if (x <= 6e-108) {
		tmp = 5.0 * y;
	} else if (x <= 3.1e+68) {
		tmp = t * x;
	} else {
		tmp = (x * y) * 2.0;
	}
	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 <= (-2.25d-48)) then
        tmp = t * x
    else if (x <= 6d-108) then
        tmp = 5.0d0 * y
    else if (x <= 3.1d+68) then
        tmp = t * x
    else
        tmp = (x * y) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.25e-48) {
		tmp = t * x;
	} else if (x <= 6e-108) {
		tmp = 5.0 * y;
	} else if (x <= 3.1e+68) {
		tmp = t * x;
	} else {
		tmp = (x * y) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.25e-48:
		tmp = t * x
	elif x <= 6e-108:
		tmp = 5.0 * y
	elif x <= 3.1e+68:
		tmp = t * x
	else:
		tmp = (x * y) * 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.25e-48)
		tmp = Float64(t * x);
	elseif (x <= 6e-108)
		tmp = Float64(5.0 * y);
	elseif (x <= 3.1e+68)
		tmp = Float64(t * x);
	else
		tmp = Float64(Float64(x * y) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.25e-48)
		tmp = t * x;
	elseif (x <= 6e-108)
		tmp = 5.0 * y;
	elseif (x <= 3.1e+68)
		tmp = t * x;
	else
		tmp = (x * y) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.25e-48], N[(t * x), $MachinePrecision], If[LessEqual[x, 6e-108], N[(5.0 * y), $MachinePrecision], If[LessEqual[x, 3.1e+68], N[(t * x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.24999999999999994e-48 or 5.99999999999999986e-108 < x < 3.0999999999999998e68

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 39.0

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

   5. Applied rewrites 39.0%

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

   if -2.24999999999999994e-48 < x < 5.99999999999999986e-108

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 60.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if 3.0999999999999998e68 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(2 \cdot y + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + 2 \cdot z\right) + 5 \cdot y} \]

      2. distribute-lft-out N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \left(y + z\right)\right)} + 5 \cdot y \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot \left(y + z\right)} + 5 \cdot y \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot x\right)} \cdot \left(y + z\right) + 5 \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, y + z, 5 \cdot y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot x}, y + z, 5 \cdot y\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot x, \color{blue}{z + y}, 5 \cdot y\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot x, \color{blue}{z + y}, 5 \cdot y\right) \]

      9. lower-*.f64 80.1

         \[\leadsto \mathsf{fma}\left(2 \cdot x, z + y, \color{blue}{5 \cdot y}\right) \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot x, z + y, 5 \cdot y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot \left(y + z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

      \[\leadsto \left(\left(z + y\right) \cdot x\right) \cdot \color{blue}{2} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(x \cdot y\right) \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 54.9%

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

Alternative 8: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+41} \lor \neg \left(y \leq 2.05 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(y + y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.25e+41) (not (<= y 2.05e+14)))
   (fma y 5.0 (* (+ y y) x))
   (* (fma 2.0 (+ z y) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e+41) || !(y <= 2.05e+14)) {
		tmp = fma(y, 5.0, ((y + y) * x));
	} else {
		tmp = fma(2.0, (z + y), t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.25e+41) || !(y <= 2.05e+14))
		tmp = fma(y, 5.0, Float64(Float64(y + y) * x));
	else
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.25e+41], N[Not[LessEqual[y, 2.05e+14]], $MachinePrecision]], N[(y * 5.0 + N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000006e41 or 2.05e14 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x\right) \]

      15. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \cdot x\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

      18. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z + y, t\right) \cdot x\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot y\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 79.3

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot y\right)} \cdot x\right) \]

   7. Applied rewrites 79.3%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot y\right)} \cdot x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 79.3%

      \[\leadsto \mathsf{fma}\left(y, 5, \left(y + \color{blue}{y}\right) \cdot x\right) \]

   if -1.25000000000000006e41 < y < 2.05e14

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 14.9

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 14.9%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 88.1

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 88.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+41} \lor \neg \left(y \leq 2.05 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(y + y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+41} \lor \neg \left(y \leq 2.05 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.25e+41) (not (<= y 2.05e+14)))
   (* (fma 2.0 x 5.0) y)
   (* (fma 2.0 (+ z y) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e+41) || !(y <= 2.05e+14)) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = fma(2.0, (z + y), t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.25e+41) || !(y <= 2.05e+14))
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.25e+41], N[Not[LessEqual[y, 2.05e+14]], $MachinePrecision]], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000006e41 or 2.05e14 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + 2 \cdot x\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + 2 \cdot x\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot x + 5\right)} \cdot y \]

      4. lower-fma.f64 79.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -1.25000000000000006e41 < y < 2.05e14

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 14.9

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 14.9%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 88.1

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 88.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+41} \lor \neg \left(y \leq 2.05 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+33} \lor \neg \left(y \leq 2.05 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -8.2e+33) (not (<= y 2.05e+14)))
   (* (fma 2.0 x 5.0) y)
   (* (fma 2.0 z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -8.2e+33) || !(y <= 2.05e+14)) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = fma(2.0, z, t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -8.2e+33) || !(y <= 2.05e+14))
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = Float64(fma(2.0, z, t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -8.2e+33], N[Not[LessEqual[y, 2.05e+14]], $MachinePrecision]], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999999e33 or 2.05e14 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + 2 \cdot x\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + 2 \cdot x\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot x + 5\right)} \cdot y \]

      4. lower-fma.f64 79.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -8.1999999999999999e33 < y < 2.05e14

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot z + t\right)} \cdot x \]

      4. lower-fma.f64 82.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x \]

   5. Applied rewrites 82.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+33} \lor \neg \left(y \leq 2.05 \cdot 10^{+14}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48} \lor \neg \left(x \leq 2.7 \cdot 10^{-125}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.25e-48) (not (<= x 2.7e-125)))
   (* (fma 2.0 y t) x)
   (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e-48) || !(x <= 2.7e-125)) {
		tmp = fma(2.0, y, t) * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.25e-48) || !(x <= 2.7e-125))
		tmp = Float64(fma(2.0, y, t) * x);
	else
		tmp = Float64(5.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.25e-48], N[Not[LessEqual[x, 2.7e-125]], $MachinePrecision]], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.24999999999999994e-48 or 2.6999999999999998e-125 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 6.4

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 6.4%

      \[\leadsto \color{blue}{5 \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(t + \color{blue}{2 \cdot \left(y + z\right)}\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot \left(y + z\right)\right) \cdot x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(y + z\right) + t\right)} \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 95.6

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(t + 2 \cdot y\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 66.3%

       \[\leadsto \mathsf{fma}\left(2, y, t\right) \cdot x \]

   if -2.24999999999999994e-48 < x < 2.6999999999999998e-125

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 61.4

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 61.4%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48} \lor \neg \left(x \leq 2.7 \cdot 10^{-125}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48} \lor \neg \left(x \leq 6 \cdot 10^{-108}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.25e-48) (not (<= x 6e-108))) (* t x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e-48) || !(x <= 6e-108)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
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 <= (-2.25d-48)) .or. (.not. (x <= 6d-108))) then
        tmp = t * x
    else
        tmp = 5.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e-48) || !(x <= 6e-108)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.25e-48) or not (x <= 6e-108):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.25e-48) || !(x <= 6e-108))
		tmp = Float64(t * x);
	else
		tmp = Float64(5.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.25e-48) || ~((x <= 6e-108)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.25e-48], N[Not[LessEqual[x, 6e-108]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.24999999999999994e-48 or 5.99999999999999986e-108 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 35.0

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

   5. Applied rewrites 35.0%

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

   if -2.24999999999999994e-48 < x < 5.99999999999999986e-108

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 60.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-48} \lor \neg \left(x \leq 6 \cdot 10^{-108}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ 5 \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 5.0 y))

C

double code(double x, double y, double z, double t) {
	return 5.0 * y;
}

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 = 5.0d0 * y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 5.0 * y;
}

Python

def code(x, y, z, t):
	return 5.0 * y

Julia

function code(x, y, z, t)
	return Float64(5.0 * y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 5.0 * y;
end

Wolfram

code[x_, y_, z_, t_] := N[(5.0 * y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 27.4

      \[\leadsto \color{blue}{5 \cdot y} \]

5. Applied rewrites 27.4%

   \[\leadsto \color{blue}{5 \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))