Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.1% → 96.9%

Time: 3.9s

Alternatives: 11

Speedup: 1.3×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + (y * z)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + (y * z)) + (t * a)) + ((a * z) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))))
   (if (<= t_1 4e+215) t_1 (fma z y (fma (fma z b t) a x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((a * z) * b);
	double tmp;
	if (t_1 <= 4e+215) {
		tmp = t_1;
	} else {
		tmp = fma(z, y, fma(fma(z, b, t), a, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
	tmp = 0.0
	if (t_1 <= 4e+215)
		tmp = t_1;
	else
		tmp = fma(z, y, fma(fma(z, b, t), a, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+215], t$95$1, N[(z * y + N[(N[(z * b + t), $MachinePrecision] * a + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < 3.99999999999999963e215

   1. Initial program 92.1%

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

   if 3.99999999999999963e215 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

Alternative 2: 95.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (fma z y (fma (fma z b t) a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(z, y, fma(fma(z, b, t), a, x));
}

Julia

function code(x, y, z, t, a, b)
	return fma(z, y, fma(fma(z, b, t), a, x))
end

Wolfram

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

Derivation

1. Initial program 92.1%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. distribute-lft-out N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 94.7

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 94.7

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

3. Applied rewrites 94.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft-out N/A

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

   13. lift-*.f64 N/A

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

   14. lift-fma.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lower-fma.f64 95.0

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

   17. lift-+.f64 N/A

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

   18. lift-fma.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. distribute-lft-out N/A

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

   21. lift-+.f64 N/A

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

   22. lift-*.f64 N/A

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

5. Applied rewrites 95.8%

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

Alternative 3: 86.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 29500:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma z b t) a x)))
   (if (<= a -6.5e+90) t_1 (if (<= a 29500.0) (fma a t (fma z y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(z, b, t), a, x);
	double tmp;
	if (a <= -6.5e+90) {
		tmp = t_1;
	} else if (a <= 29500.0) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(z, b, t), a, x)
	tmp = 0.0
	if (a <= -6.5e+90)
		tmp = t_1;
	elseif (a <= 29500.0)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * b + t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[a, -6.5e+90], t$95$1, If[LessEqual[a, 29500.0], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000001e90 or 29500 < a

   1. Initial program 92.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 74.2

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

   4. Applied rewrites 74.2%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 75.0

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

   6. Applied rewrites 75.0%

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

   if -6.5000000000000001e90 < a < 29500

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 77.3%

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

Alternative 4: 82.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma z b t) a x)))
   (if (<= a -1.5e-29) t_1 (if (<= a 1.9e-52) (fma y z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(z, b, t), a, x);
	double tmp;
	if (a <= -1.5e-29) {
		tmp = t_1;
	} else if (a <= 1.9e-52) {
		tmp = fma(y, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(z, b, t), a, x)
	tmp = 0.0
	if (a <= -1.5e-29)
		tmp = t_1;
	elseif (a <= 1.9e-52)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * b + t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[a, -1.5e-29], t$95$1, If[LessEqual[a, 1.9e-52], N[(y * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5000000000000001e-29 or 1.9000000000000002e-52 < a

   1. Initial program 92.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 74.2

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

   4. Applied rewrites 74.2%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 75.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 75.0

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

   6. Applied rewrites 75.0%

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

   if -1.5000000000000001e-29 < a < 1.9000000000000002e-52

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

      1. +-commutative 51.6

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

      2. lift-fma.f64 N/A

         \[\leadsto x + y \cdot z \]

      3. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      6. *-commutative N/A

         \[\leadsto x + y \cdot z \]

      7. lift-fma.f64 51.6

         \[\leadsto x + y \cdot z \]

      8. distribute-lft-in 51.6

         \[\leadsto x + y \cdot z \]

      9. associate-*l* 51.6

         \[\leadsto x + y \cdot z \]

      10. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      11. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      12. lift-*.f64 N/A

          \[\leadsto x + y \cdot z \]

      13. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      14. associate-+r+ N/A

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

      15. +-commutative N/A

          \[\leadsto x + y \cdot z \]

      16. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      17. lift-*.f64 51.6

          \[\leadsto x + y \cdot z \]

      18. +-commutative 51.6

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

      19. lift-+.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   10. Applied rewrites 51.6%

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

Alternative 5: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b, t\right) \cdot a\\ \mathbf{if}\;a \leq -1.56 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{+83}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma z b t) a)))
   (if (<= a -1.56e+21)
     t_1
     (if (<= a 4.6e-48) (fma y z x) (if (<= a 7.6e+83) (+ x (* a t)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, b, t) * a;
	double tmp;
	if (a <= -1.56e+21) {
		tmp = t_1;
	} else if (a <= 4.6e-48) {
		tmp = fma(y, z, x);
	} else if (a <= 7.6e+83) {
		tmp = x + (a * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(z, b, t) * a)
	tmp = 0.0
	if (a <= -1.56e+21)
		tmp = t_1;
	elseif (a <= 4.6e-48)
		tmp = fma(y, z, x);
	elseif (a <= 7.6e+83)
		tmp = Float64(x + Float64(a * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * b + t), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.56e+21], t$95$1, If[LessEqual[a, 4.6e-48], N[(y * z + x), $MachinePrecision], If[LessEqual[a, 7.6e+83], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.56e21 or 7.6000000000000004e83 < a

   1. Initial program 92.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 51.6

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

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.6

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

      4. lift-+.f64 N/A

         \[\leadsto \left(t + b \cdot z\right) \cdot a \]

      5. +-commutative N/A

         \[\leadsto \left(b \cdot z + t\right) \cdot a \]

      6. lift-*.f64 N/A

         \[\leadsto \left(b \cdot z + t\right) \cdot a \]

      7. *-commutative N/A

         \[\leadsto \left(z \cdot b + t\right) \cdot a \]

      8. lower-fma.f64 51.6

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

   6. Applied rewrites 51.6%

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

   if -1.56e21 < a < 4.6000000000000001e-48

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

      1. +-commutative 51.6

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

      2. lift-fma.f64 N/A

         \[\leadsto x + y \cdot z \]

      3. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      6. *-commutative N/A

         \[\leadsto x + y \cdot z \]

      7. lift-fma.f64 51.6

         \[\leadsto x + y \cdot z \]

      8. distribute-lft-in 51.6

         \[\leadsto x + y \cdot z \]

      9. associate-*l* 51.6

         \[\leadsto x + y \cdot z \]

      10. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      11. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      12. lift-*.f64 N/A

          \[\leadsto x + y \cdot z \]

      13. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      14. associate-+r+ N/A

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

      15. +-commutative N/A

          \[\leadsto x + y \cdot z \]

      16. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      17. lift-*.f64 51.6

          \[\leadsto x + y \cdot z \]

      18. +-commutative 51.6

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

      19. lift-+.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   10. Applied rewrites 51.6%

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

   if 4.6000000000000001e-48 < a < 7.6000000000000004e83

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

      \[\leadsto \color{blue}{x + a \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.004:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -2.4e-5) t_1 (if (<= z 0.004) (+ x (* a t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, y) * z;
	double tmp;
	if (z <= -2.4e-5) {
		tmp = t_1;
	} else if (z <= 0.004) {
		tmp = x + (a * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -2.4e-5)
		tmp = t_1;
	elseif (z <= 0.004)
		tmp = Float64(x + Float64(a * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.4e-5], t$95$1, If[LessEqual[z, 0.004], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4000000000000001e-5 or 0.0040000000000000001 < z

   1. Initial program 92.1%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.7

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

      4. lift-+.f64 N/A

         \[\leadsto \left(y + a \cdot b\right) \cdot z \]

      5. +-commutative N/A

         \[\leadsto \left(a \cdot b + y\right) \cdot z \]

      6. lift-*.f64 N/A

         \[\leadsto \left(a \cdot b + y\right) \cdot z \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot a + y\right) \cdot z \]

      8. lower-fma.f64 50.7

         \[\leadsto \mathsf{fma}\left(b, a, y\right) \cdot z \]

   6. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, y\right) \cdot z} \]

   if -2.4000000000000001e-5 < z < 0.0040000000000000001

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 7: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;a \leq -5.9 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6.5 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot z, a, x\right)\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a t))))
   (if (<= a -5.9e+125)
     t_1
     (if (<= a -6.5e+90)
       (fma (* b z) a x)
       (if (<= a 4.6e-48) (fma y z x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * t);
	double tmp;
	if (a <= -5.9e+125) {
		tmp = t_1;
	} else if (a <= -6.5e+90) {
		tmp = fma((b * z), a, x);
	} else if (a <= 4.6e-48) {
		tmp = fma(y, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(a * t))
	tmp = 0.0
	if (a <= -5.9e+125)
		tmp = t_1;
	elseif (a <= -6.5e+90)
		tmp = fma(Float64(b * z), a, x);
	elseif (a <= 4.6e-48)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.9e+125], t$95$1, If[LessEqual[a, -6.5e+90], N[(N[(b * z), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[a, 4.6e-48], N[(y * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9000000000000001e125 or 4.6000000000000001e-48 < a

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if -5.9000000000000001e125 < a < -6.5000000000000001e90

   1. Initial program 92.1%

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

   2. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 50.4

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

   7. Applied rewrites 50.4%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.4

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

   9. Applied rewrites 50.4%

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

   if -6.5000000000000001e90 < a < 4.6000000000000001e-48

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

      1. +-commutative 51.6

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

      2. lift-fma.f64 N/A

         \[\leadsto x + y \cdot z \]

      3. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      6. *-commutative N/A

         \[\leadsto x + y \cdot z \]

      7. lift-fma.f64 51.6

         \[\leadsto x + y \cdot z \]

      8. distribute-lft-in 51.6

         \[\leadsto x + y \cdot z \]

      9. associate-*l* 51.6

         \[\leadsto x + y \cdot z \]

      10. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      11. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      12. lift-*.f64 N/A

          \[\leadsto x + y \cdot z \]

      13. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      14. associate-+r+ N/A

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

      15. +-commutative N/A

          \[\leadsto x + y \cdot z \]

      16. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      17. lift-*.f64 51.6

          \[\leadsto x + y \cdot z \]

      18. +-commutative 51.6

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

      19. lift-+.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   10. Applied rewrites 51.6%

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

Alternative 8: 62.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + a \cdot t\\ \mathbf{if}\;a \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* a t))))
   (if (<= a -1.65e+21) t_1 (if (<= a 4.6e-48) (fma y z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (a * t);
	double tmp;
	if (a <= -1.65e+21) {
		tmp = t_1;
	} else if (a <= 4.6e-48) {
		tmp = fma(y, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(a * t))
	tmp = 0.0
	if (a <= -1.65e+21)
		tmp = t_1;
	elseif (a <= 4.6e-48)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.65e+21], t$95$1, If[LessEqual[a, 4.6e-48], N[(y * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.65e21 or 4.6000000000000001e-48 < a

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if -1.65e21 < a < 4.6000000000000001e-48

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

      1. +-commutative 51.6

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

      2. lift-fma.f64 N/A

         \[\leadsto x + y \cdot z \]

      3. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      6. *-commutative N/A

         \[\leadsto x + y \cdot z \]

      7. lift-fma.f64 51.6

         \[\leadsto x + y \cdot z \]

      8. distribute-lft-in 51.6

         \[\leadsto x + y \cdot z \]

      9. associate-*l* 51.6

         \[\leadsto x + y \cdot z \]

      10. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      11. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      12. lift-*.f64 N/A

          \[\leadsto x + y \cdot z \]

      13. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      14. associate-+r+ N/A

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

      15. +-commutative N/A

          \[\leadsto x + y \cdot z \]

      16. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      17. lift-*.f64 51.6

          \[\leadsto x + y \cdot z \]

      18. +-commutative 51.6

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

      19. lift-+.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   10. Applied rewrites 51.6%

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

Alternative 9: 57.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -7.5e-15) (* a t) (if (<= t 2.4e+122) (fma y z x) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7.5e-15) {
		tmp = a * t;
	} else if (t <= 2.4e+122) {
		tmp = fma(y, z, x);
	} else {
		tmp = a * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -7.5e-15)
		tmp = Float64(a * t);
	elseif (t <= 2.4e+122)
		tmp = fma(y, z, x);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -7.5e-15], N[(a * t), $MachinePrecision], If[LessEqual[t, 2.4e+122], N[(y * z + x), $MachinePrecision], N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.4999999999999996e-15 or 2.4000000000000002e122 < t

   1. Initial program 92.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto a \cdot t \]
   6. Step-by-step derivation

   7. Applied rewrites 28.7%

      \[\leadsto a \cdot t \]

   if -7.4999999999999996e-15 < t < 2.4000000000000002e122

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

      1. +-commutative 51.6

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

      2. lift-fma.f64 N/A

         \[\leadsto x + y \cdot z \]

      3. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

         \[\leadsto x + y \cdot z \]

      6. *-commutative N/A

         \[\leadsto x + y \cdot z \]

      7. lift-fma.f64 51.6

         \[\leadsto x + y \cdot z \]

      8. distribute-lft-in 51.6

         \[\leadsto x + y \cdot z \]

      9. associate-*l* 51.6

         \[\leadsto x + y \cdot z \]

      10. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      11. *-commutative 51.6

          \[\leadsto x + y \cdot z \]

      12. lift-*.f64 N/A

          \[\leadsto x + y \cdot z \]

      13. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      14. associate-+r+ N/A

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

      15. +-commutative N/A

          \[\leadsto x + y \cdot z \]

      16. lift-+.f64 N/A

          \[\leadsto x + y \cdot z \]

      17. lift-*.f64 51.6

          \[\leadsto x + y \cdot z \]

      18. +-commutative 51.6

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

      19. lift-+.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   10. Applied rewrites 51.6%

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

Alternative 10: 39.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{+20}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 5.7 \cdot 10^{-43}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.9e+20) (* a t) (if (<= a 5.7e-43) (* y z) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.9e+20) {
		tmp = a * t;
	} else if (a <= 5.7e-43) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.9d+20)) then
        tmp = a * t
    else if (a <= 5.7d-43) then
        tmp = y * z
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.9e+20) {
		tmp = a * t;
	} else if (a <= 5.7e-43) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -3.9e+20:
		tmp = a * t
	elif a <= 5.7e-43:
		tmp = y * z
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.9e+20)
		tmp = Float64(a * t);
	elseif (a <= 5.7e-43)
		tmp = Float64(y * z);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -3.9e+20)
		tmp = a * t;
	elseif (a <= 5.7e-43)
		tmp = y * z;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.9e+20], N[(a * t), $MachinePrecision], If[LessEqual[a, 5.7e-43], N[(y * z), $MachinePrecision], N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9e20 or 5.6999999999999999e-43 < a

   1. Initial program 92.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto a \cdot t \]
   6. Step-by-step derivation

   7. Applied rewrites 28.7%

      \[\leadsto a \cdot t \]

   if -3.9e20 < a < 5.6999999999999999e-43

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.7

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

   3. Applied rewrites 94.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-fma.f64 95.0

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

      17. lift-+.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. distribute-lft-out N/A

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

      21. lift-+.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 27.8

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

   8. Applied rewrites 27.8%

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

Alternative 11: 27.8% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (* y z))

C

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

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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = y * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. distribute-lft-out N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 94.7

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 94.7

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

3. Applied rewrites 94.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-lft-out N/A

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

   13. lift-*.f64 N/A

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

   14. lift-fma.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lower-fma.f64 95.0

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

   17. lift-+.f64 N/A

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

   18. lift-fma.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. distribute-lft-out N/A

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

   21. lift-+.f64 N/A

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

   22. lift-*.f64 N/A

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

5. Applied rewrites 95.8%

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

6. Taylor expanded in y around inf

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

   1. lower-*.f64 27.8

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

8. Applied rewrites 27.8%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64
  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))