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

Percentage Accurate: 99.9% → 100.0%

Time: 3.4s

Alternatives: 12

Speedup: 1.1×

• 26.8% 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. 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. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(z, 2, t\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -2.5) t_1 (if (<= x 2.5) (fma y 5.0 (* (fma z 2.0 t) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -2.5) {
		tmp = t_1;
	} else if (x <= 2.5) {
		tmp = fma(y, 5.0, (fma(z, 2.0, t) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_1;
	elseif (x <= 2.5)
		tmp = fma(y, 5.0, Float64(fma(z, 2.0, t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$1, If[LessEqual[x, 2.5], N[(y * 5.0 + N[(N[(z * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 2.5 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. 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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. count-2-rev N/A

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

       4. associate-+l+ N/A

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

       5. lower-*.f64 N/A

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

       6. associate-+l+ N/A

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

       7. count-2-rev N/A

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

       8. +-commutative N/A

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

       9. distribute-lft-out N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if -2.5 < 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. 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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 84.8

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

   6. Applied rewrites 84.8%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(2.0 * x + 5.0), $MachinePrecision] * y + N[(N[(2.0 * z + 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. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 98.5

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

4. Applied rewrites 98.5%

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

Alternative 4: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \mathbf{if}\;y \leq -0.0245:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (fma 2.0 y t) x (* 5.0 y))))
   (if (<= y -0.0245) t_1 (if (<= y 1.15e+26) (* x (fma 2.0 (+ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(fma(2.0, y, t), x, (5.0 * y));
	double tmp;
	if (y <= -0.0245) {
		tmp = t_1;
	} else if (y <= 1.15e+26) {
		tmp = x * fma(2.0, (y + z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(fma(2.0, y, t), x, Float64(5.0 * y))
	tmp = 0.0
	if (y <= -0.0245)
		tmp = t_1;
	elseif (y <= 1.15e+26)
		tmp = Float64(x * fma(2.0, Float64(y + z), t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * y + t), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0245], t$95$1, If[LessEqual[y, 1.15e+26], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.024500000000000001 or 1.15e26 < 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. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

   if -0.024500000000000001 < y < 1.15e26

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. 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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. count-2-rev N/A

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

       4. associate-+l+ N/A

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

       5. lower-*.f64 N/A

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

       6. associate-+l+ N/A

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

       7. count-2-rev N/A

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

       8. +-commutative N/A

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

       9. distribute-lft-out N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 71.8

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

   11. Applied rewrites 71.8%

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

Alternative 5: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x, 2, 5\right), y, x \cdot t\right)\\ \mathbf{if}\;y \leq -0.0245:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (fma x 2.0 5.0) y (* x t))))
   (if (<= y -0.0245) t_1 (if (<= y 1.15e+26) (* x (fma 2.0 (+ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(fma(x, 2.0, 5.0), y, (x * t));
	double tmp;
	if (y <= -0.0245) {
		tmp = t_1;
	} else if (y <= 1.15e+26) {
		tmp = x * fma(2.0, (y + z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(fma(x, 2.0, 5.0), y, Float64(x * t))
	tmp = 0.0
	if (y <= -0.0245)
		tmp = t_1;
	elseif (y <= 1.15e+26)
		tmp = Float64(x * fma(2.0, Float64(y + z), t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 2.0 + 5.0), $MachinePrecision] * y + N[(x * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0245], t$95$1, If[LessEqual[y, 1.15e+26], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.024500000000000001 or 1.15e26 < 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. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

   if -0.024500000000000001 < y < 1.15e26

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. 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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. count-2-rev N/A

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

       4. associate-+l+ N/A

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

       5. lower-*.f64 N/A

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

       6. associate-+l+ N/A

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

       7. count-2-rev N/A

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

       8. +-commutative N/A

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

       9. distribute-lft-out N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 71.8

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

   11. Applied rewrites 71.8%

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

Alternative 6: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -5.5e-7) t_1 (if (<= x 1.75e-10) (fma y 5.0 (* t x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -5.5e-7) {
		tmp = t_1;
	} else if (x <= 1.75e-10) {
		tmp = fma(y, 5.0, (t * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -5.5e-7)
		tmp = t_1;
	elseif (x <= 1.75e-10)
		tmp = fma(y, 5.0, Float64(t * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-7], t$95$1, If[LessEqual[x, 1.75e-10], N[(y * 5.0 + N[(t * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000003e-7 or 1.7499999999999999e-10 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. 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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. count-2-rev N/A

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

       4. associate-+l+ N/A

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

       5. lower-*.f64 N/A

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

       6. associate-+l+ N/A

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

       7. count-2-rev N/A

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

       8. +-commutative N/A

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

       9. distribute-lft-out N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if -5.5000000000000003e-7 < x < 1.7499999999999999e-10

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 58.0

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

   9. Applied rewrites 58.0%

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

Alternative 7: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -1.45e+29) t_1 (if (<= y 4.4e+57) (* (fma 2.0 z 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 <= -1.45e+29) {
		tmp = t_1;
	} else if (y <= 4.4e+57) {
		tmp = fma(2.0, z, 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 <= -1.45e+29)
		tmp = t_1;
	elseif (y <= 4.4e+57)
		tmp = Float64(fma(2.0, z, 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, -1.45e+29], t$95$1, If[LessEqual[y, 4.4e+57], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e29 or 4.4000000000000001e57 < 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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   3. 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 47.6

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

   4. Applied rewrites 47.6%

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

   if -1.45e29 < y < 4.4000000000000001e57

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   3. 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 57.1

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

   4. Applied rewrites 57.1%

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

Alternative 8: 59.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, y, t\right) \cdot x\\ t_2 := \left(x + x\right) \cdot z\\ t_3 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+57}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-219}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4000000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 y t) x))
        (t_2 (* (+ x x) z))
        (t_3 (* (fma 2.0 x 5.0) y)))
   (if (<= y -7.5e+57)
     t_3
     (if (<= y -1.15e-156)
       t_1
       (if (<= y 1.02e-219)
         t_2
         (if (<= y 8.2e-129) t_1 (if (<= y 4000000000.0) t_2 t_3)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, y, t) * x;
	double t_2 = (x + x) * z;
	double t_3 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -7.5e+57) {
		tmp = t_3;
	} else if (y <= -1.15e-156) {
		tmp = t_1;
	} else if (y <= 1.02e-219) {
		tmp = t_2;
	} else if (y <= 8.2e-129) {
		tmp = t_1;
	} else if (y <= 4000000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, y, t) * x)
	t_2 = Float64(Float64(x + x) * z)
	t_3 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -7.5e+57)
		tmp = t_3;
	elseif (y <= -1.15e-156)
		tmp = t_1;
	elseif (y <= 1.02e-219)
		tmp = t_2;
	elseif (y <= 8.2e-129)
		tmp = t_1;
	elseif (y <= 4000000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -7.5e+57], t$95$3, If[LessEqual[y, -1.15e-156], t$95$1, If[LessEqual[y, 1.02e-219], t$95$2, If[LessEqual[y, 8.2e-129], t$95$1, If[LessEqual[y, 4000000000.0], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5000000000000006e57 or 4e9 < 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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   3. 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 47.6

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

   4. Applied rewrites 47.6%

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

   if -7.5000000000000006e57 < y < -1.15e-156 or 1.02e-219 < y < 8.1999999999999999e-129

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

   if -1.15e-156 < y < 1.02e-219 or 8.1999999999999999e-129 < y < 4e9

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

Alternative 9: 57.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + x\right) \cdot z\\ \mathbf{if}\;z \leq -9 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.062:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ x x) z)))
   (if (<= z -9e+77) t_1 (if (<= z 0.062) (* (fma 2.0 y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + x) * z;
	double tmp;
	if (z <= -9e+77) {
		tmp = t_1;
	} else if (z <= 0.062) {
		tmp = fma(2.0, y, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + x) * z)
	tmp = 0.0
	if (z <= -9e+77)
		tmp = t_1;
	elseif (z <= 0.062)
		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[(x + x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -9e+77], t$95$1, If[LessEqual[z, 0.062], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.00000000000000049e77 or 0.062 < z

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

   if -9.00000000000000049e77 < z < 0.062

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 46.4

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

   7. Applied rewrites 46.4%

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

Alternative 10: 47.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+65}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;t \leq 13500000000000:\\ \;\;\;\;\left(x + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.85e+65)
   (* t x)
   (if (<= t 13500000000000.0) (* (+ x x) z) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.85e+65) {
		tmp = t * x;
	} else if (t <= 13500000000000.0) {
		tmp = (x + x) * z;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.85d+65)) then
        tmp = t * x
    else if (t <= 13500000000000.0d0) then
        tmp = (x + x) * z
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.85e+65) {
		tmp = t * x;
	} else if (t <= 13500000000000.0) {
		tmp = (x + x) * z;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.85e+65:
		tmp = t * x
	elif t <= 13500000000000.0:
		tmp = (x + x) * z
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.85e+65)
		tmp = Float64(t * x);
	elseif (t <= 13500000000000.0)
		tmp = Float64(Float64(x + x) * z);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.85e+65)
		tmp = t * x;
	elseif (t <= 13500000000000.0)
		tmp = (x + x) * z;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.85e+65], N[(t * x), $MachinePrecision], If[LessEqual[t, 13500000000000.0], N[(N[(x + x), $MachinePrecision] * z), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.84999999999999997e65 or 1.35e13 < t

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 30.5

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

   4. Applied rewrites 30.5%

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

   if -1.84999999999999997e65 < t < 1.35e13

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

Alternative 11: 46.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-7}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-62}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.5e-7) (* t x) (if (<= x 5.3e-62) (* 5.0 y) (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.5e-7) {
		tmp = t * x;
	} else if (x <= 5.3e-62) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-6.5d-7)) then
        tmp = t * x
    else if (x <= 5.3d-62) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.5e-7) {
		tmp = t * x;
	} else if (x <= 5.3e-62) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.5e-7:
		tmp = t * x
	elif x <= 5.3e-62:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.5e-7)
		tmp = Float64(t * x);
	elseif (x <= 5.3e-62)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6.5e-7)
		tmp = t * x;
	elseif (x <= 5.3e-62)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.5e-7], N[(t * x), $MachinePrecision], If[LessEqual[x, 5.3e-62], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.50000000000000024e-7 or 5.2999999999999997e-62 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 30.5

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

   4. Applied rewrites 30.5%

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

   if -6.50000000000000024e-7 < x < 5.2999999999999997e-62

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 29.9

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

   4. Applied rewrites 29.9%

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

Alternative 12: 29.9% accurate, 5.2× 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. Taylor expanded in x around 0

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

   1. lower-*.f64 29.9

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

4. Applied rewrites 29.9%

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

Reproduce

herbie shell --seed 2025127 
(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)))