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

Percentage Accurate: 99.9% → 100.0%

Time: 3.7s

Alternatives: 9

Speedup: 1.1×

• 26.6% 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: 99.9% 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: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(N[(N[(N[(2.0 * z + y), $MachinePrecision] + 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. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 45.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + x\right) \cdot y\\ \mathbf{if}\;x \leq -7 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-103}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-126}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+184}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ x x) y)))
   (if (<= x -7e+49)
     t_1
     (if (<= x -1.8e-103)
       (* t x)
       (if (<= x 8.5e-126) (* 5.0 y) (if (<= x 8e+184) (* t x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + x) * y;
	double tmp;
	if (x <= -7e+49) {
		tmp = t_1;
	} else if (x <= -1.8e-103) {
		tmp = t * x;
	} else if (x <= 8.5e-126) {
		tmp = 5.0 * y;
	} else if (x <= 8e+184) {
		tmp = t * 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 = (x + x) * y
    if (x <= (-7d+49)) then
        tmp = t_1
    else if (x <= (-1.8d-103)) then
        tmp = t * x
    else if (x <= 8.5d-126) then
        tmp = 5.0d0 * y
    else if (x <= 8d+184) then
        tmp = t * 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 = (x + x) * y;
	double tmp;
	if (x <= -7e+49) {
		tmp = t_1;
	} else if (x <= -1.8e-103) {
		tmp = t * x;
	} else if (x <= 8.5e-126) {
		tmp = 5.0 * y;
	} else if (x <= 8e+184) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + x) * y
	tmp = 0
	if x <= -7e+49:
		tmp = t_1
	elif x <= -1.8e-103:
		tmp = t * x
	elif x <= 8.5e-126:
		tmp = 5.0 * y
	elif x <= 8e+184:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + x) * y)
	tmp = 0.0
	if (x <= -7e+49)
		tmp = t_1;
	elseif (x <= -1.8e-103)
		tmp = Float64(t * x);
	elseif (x <= 8.5e-126)
		tmp = Float64(5.0 * y);
	elseif (x <= 8e+184)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + x) * y;
	tmp = 0.0;
	if (x <= -7e+49)
		tmp = t_1;
	elseif (x <= -1.8e-103)
		tmp = t * x;
	elseif (x <= 8.5e-126)
		tmp = 5.0 * y;
	elseif (x <= 8e+184)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, -7e+49], t$95$1, If[LessEqual[x, -1.8e-103], N[(t * x), $MachinePrecision], If[LessEqual[x, 8.5e-126], N[(5.0 * y), $MachinePrecision], If[LessEqual[x, 8e+184], N[(t * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.9999999999999995e49 or 8.00000000000000014e184 < 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. 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. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      4. +-commutative N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      5. count-2-rev N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      6. associate-+l+ N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      7. associate-+l+ N/A

         \[\leadsto y \cdot \left(5 + 2 \cdot x\right) \]

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

          \[\leadsto \left(2 \cdot x + 5\right) \cdot y \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot 2 + 5\right) \cdot y \]

      12. lower-fma.f64 50.9

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

   7. Applied rewrites 50.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 50.9

         \[\leadsto \left(2 \cdot x\right) \cdot y \]

   10. Applied rewrites 50.9%

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

       1. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot x\right) \cdot y \]

       2. count-2-rev N/A

          \[\leadsto \left(x + x\right) \cdot y \]

       3. lower-+.f64 50.9

          \[\leadsto \left(x + x\right) \cdot y \]

   12. Applied rewrites 50.9%

       \[\leadsto \left(x + x\right) \cdot y \]

   if -6.9999999999999995e49 < x < -1.7999999999999999e-103 or 8.49999999999999938e-126 < x < 8.00000000000000014e184

   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -1.7999999999999999e-103 < x < 8.49999999999999938e-126

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

Alternative 3: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + y\right) \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.00105)
   (* (fma 2.0 y t) x)
   (if (<= x 8.6e-30)
     (fma y 5.0 (* t x))
     (if (<= x 2.5e+184) (* (fma 2.0 z t) x) (* (* (+ z y) x) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.00105) {
		tmp = fma(2.0, y, t) * x;
	} else if (x <= 8.6e-30) {
		tmp = fma(y, 5.0, (t * x));
	} else if (x <= 2.5e+184) {
		tmp = fma(2.0, z, t) * x;
	} else {
		tmp = ((z + y) * x) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.00105)
		tmp = Float64(fma(2.0, y, t) * x);
	elseif (x <= 8.6e-30)
		tmp = fma(y, 5.0, Float64(t * x));
	elseif (x <= 2.5e+184)
		tmp = Float64(fma(2.0, z, t) * x);
	else
		tmp = Float64(Float64(Float64(z + y) * x) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.00105], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 8.6e-30], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e+184], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.00104999999999999994

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(2, y, t\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 75.1%

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

   if -0.00104999999999999994 < x < 8.59999999999999932e-30

   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. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-+l+ 88.8

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

      2. count-2-rev 88.8

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

      3. distribute-lft-in 88.8

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

   7. Applied rewrites 88.8%

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

   if 8.59999999999999932e-30 < x < 2.4999999999999999e184

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot z + t\right) \cdot x \]

      4. lower-fma.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if 2.4999999999999999e184 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y + z\right)\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y + z\right)\right) \cdot 2 \]

      3. *-commutative N/A

         \[\leadsto \left(\left(y + z\right) \cdot x\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(y + z\right) \cdot x\right) \cdot 2 \]

      5. +-commutative N/A

         \[\leadsto \left(\left(z + y\right) \cdot x\right) \cdot 2 \]

      6. lift-+.f64 84.0

         \[\leadsto \left(\left(z + y\right) \cdot x\right) \cdot 2 \]

   8. Applied rewrites 84.0%

      \[\leadsto \left(\left(z + y\right) \cdot x\right) \cdot \color{blue}{2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-16} \lor \neg \left(x \leq 7.5 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.5e-16) (not (<= x 7.5e-30)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-16) || !(x <= 7.5e-30)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, (t * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.5e-16) || !(x <= 7.5e-30))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(t * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.5e-16], N[Not[LessEqual[x, 7.5e-30]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999997e-16 or 7.5000000000000006e-30 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -1.49999999999999997e-16 < x < 7.5000000000000006e-30

   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. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-+l+ 89.9

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

      2. count-2-rev 89.9

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

      3. distribute-lft-in 89.9

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

   7. Applied rewrites 89.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-16} \lor \neg \left(x \leq 7.5 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00105 \lor \neg \left(x \leq 5.5 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.00105) (not (<= x 5.5e-11)))
   (* (fma 2.0 y t) x)
   (fma y 5.0 (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.00105) || !(x <= 5.5e-11)) {
		tmp = fma(2.0, y, t) * x;
	} else {
		tmp = fma(y, 5.0, (t * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.00105) || !(x <= 5.5e-11))
		tmp = Float64(fma(2.0, y, t) * x);
	else
		tmp = fma(y, 5.0, Float64(t * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.00105], N[Not[LessEqual[x, 5.5e-11]], $MachinePrecision]], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00104999999999999994 or 5.49999999999999975e-11 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot y + 2 \cdot z\right) + t\right) \cdot x \]

      4. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(y + z\right) + t\right) \cdot x \]

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(2, y, t\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 72.5%

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

   if -0.00104999999999999994 < x < 5.49999999999999975e-11

   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. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-+l+ 86.8

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

      2. count-2-rev 86.8

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

      3. distribute-lft-in 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105 \lor \neg \left(x \leq 5.5 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+47} \lor \neg \left(y \leq 1.7 \cdot 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.15e+47) (not (<= y 1.7e-35)))
   (* (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 <= -3.15e+47) || !(y <= 1.7e-35)) {
		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 <= -3.15e+47) || !(y <= 1.7e-35))
		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, -3.15e+47], N[Not[LessEqual[y, 1.7e-35]], $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 < -3.15000000000000002e47 or 1.7000000000000001e-35 < 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot x + 5\right) \cdot y \]

      4. lower-fma.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -3.15000000000000002e47 < y < 1.7000000000000001e-35

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot z + t\right) \cdot x \]

      4. lower-fma.f64 82.6

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

   5. Applied rewrites 82.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{+47} \lor \neg \left(y \leq 1.7 \cdot 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-29} \lor \neg \left(y \leq 4.1 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.15e-29) (not (<= y 4.1e-37))) (* (fma 2.0 x 5.0) y) (* t x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.15e-29) || !(y <= 4.1e-37)) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.15e-29) || !(y <= 4.1e-37))
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.15e-29], N[Not[LessEqual[y, 4.1e-37]], $MachinePrecision]], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], N[(t * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1499999999999999e-29 or 4.0999999999999998e-37 < 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(2 \cdot x + 5\right) \cdot y \]

      4. lower-fma.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -2.1499999999999999e-29 < y < 4.0999999999999998e-37

   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 52.1

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

   5. Applied rewrites 52.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-29} \lor \neg \left(y \leq 4.1 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103} \lor \neg \left(x \leq 8.5 \cdot 10^{-126}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.8e-103) (not (<= x 8.5e-126))) (* t x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.8e-103) || !(x <= 8.5e-126)) {
		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 <= (-1.8d-103)) .or. (.not. (x <= 8.5d-126))) 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 <= -1.8e-103) || !(x <= 8.5e-126)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.8e-103) or not (x <= 8.5e-126):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.8e-103) || !(x <= 8.5e-126))
		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 <= -1.8e-103) || ~((x <= 8.5e-126)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.8e-103], N[Not[LessEqual[x, 8.5e-126]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7999999999999999e-103 or 8.49999999999999938e-126 < 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 39.6

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

   5. Applied rewrites 39.6%

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

   if -1.7999999999999999e-103 < x < 8.49999999999999938e-126

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-103} \lor \neg \left(x \leq 8.5 \cdot 10^{-126}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.1% 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 30.4

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

5. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2025051 
(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)))