Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.1% → 94.8%

Time: 3.4s

Alternatives: 13

Speedup: 1.3×

• 68.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y + 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. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. remove-double-neg N/A

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

   15. remove-double-neg N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 94.8%

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. add-flip N/A

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

   21. sub-flip N/A

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. remove-double-neg N/A

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

   25. lower-fma.f64 94.8%

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

3. Applied rewrites 94.8%

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

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x + \mathsf{fma}\left(a, t, a \cdot \left(b \cdot z\right)\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + a \cdot t\right) + \left(a \cdot z\right) \cdot b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.85e+101)
   (+ x (fma a t (* a (* b z))))
   (if (<= b 5.2e-65) (fma t a (fma z y x)) (+ (+ x (* a t)) (* (* a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.85e+101) {
		tmp = x + fma(a, t, (a * (b * z)));
	} else if (b <= 5.2e-65) {
		tmp = fma(t, a, fma(z, y, x));
	} else {
		tmp = (x + (a * t)) + ((a * z) * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.85e+101)
		tmp = Float64(x + fma(a, t, Float64(a * Float64(b * z))));
	elseif (b <= 5.2e-65)
		tmp = fma(t, a, fma(z, y, x));
	else
		tmp = Float64(Float64(x + Float64(a * t)) + Float64(Float64(a * z) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.85e+101], N[(x + N[(a * t + N[(a * N[(b * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-65], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8499999999999999e101

   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 74.6%

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

   4. Applied rewrites 74.6%

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

   if -1.8499999999999999e101 < b < 5.2000000000000002e-65

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.5%

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

   if 5.2000000000000002e-65 < 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. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 72.2%

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

   4. Applied rewrites 72.2%

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

Alternative 3: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := x + \mathsf{fma}\left(a, t, a \cdot \left(b \cdot z\right)\right)\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma a t (* a (* b z))))))
   (if (<= b -1.85e+101) t_1 (if (<= b 2.7e-82) (fma t a (fma z y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(a, t, (a * (b * z)));
	double tmp;
	if (b <= -1.85e+101) {
		tmp = t_1;
	} else if (b <= 2.7e-82) {
		tmp = fma(t, a, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(a, t, Float64(a * Float64(b * z))))
	tmp = 0.0
	if (b <= -1.85e+101)
		tmp = t_1;
	elseif (b <= 2.7e-82)
		tmp = fma(t, a, 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[(x + N[(a * t + N[(a * N[(b * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.85e+101], t$95$1, If[LessEqual[b, 2.7e-82], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.8499999999999999e101 or 2.7000000000000001e-82 < 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. 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 74.6%

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

   4. Applied rewrites 74.6%

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

   if -1.8499999999999999e101 < b < 2.7000000000000001e-82

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.5%

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

Alternative 4: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(z, b, t\right) \cdot a\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot b, a, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.2e+236)
   (* (fma z b t) a)
   (if (<= b 8.6e+189) (fma t a (fma z y x)) (fma (* z b) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.2e+236) {
		tmp = fma(z, b, t) * a;
	} else if (b <= 8.6e+189) {
		tmp = fma(t, a, fma(z, y, x));
	} else {
		tmp = fma((z * b), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.2e+236)
		tmp = Float64(fma(z, b, t) * a);
	elseif (b <= 8.6e+189)
		tmp = fma(t, a, fma(z, y, x));
	else
		tmp = fma(Float64(z * b), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.2e+236], N[(N[(z * b + t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 8.6e+189], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(z * b), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2000000000000001e236

   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 51.9%

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

   4. Applied rewrites 51.9%

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   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 51.9%

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

      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 51.9%

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

   6. Applied rewrites 51.9%

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

   if -1.2000000000000001e236 < b < 8.6e189

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.5%

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

   if 8.6e189 < 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. 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 70.3%

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

   4. Applied rewrites 70.3%

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

   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 50.7%

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

   7. Applied rewrites 50.7%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z\right)} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.7%

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

   9. Applied rewrites 50.7%

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

Alternative 5: 66.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right) \cdot z\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot b, a, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.65e-71)
   (* (fma a b y) z)
   (if (<= z 2.1e-21) (fma a t x) (fma (* z b) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.65e-71) {
		tmp = fma(a, b, y) * z;
	} else if (z <= 2.1e-21) {
		tmp = fma(a, t, x);
	} else {
		tmp = fma((z * b), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.65e-71)
		tmp = Float64(fma(a, b, y) * z);
	elseif (z <= 2.1e-21)
		tmp = fma(a, t, x);
	else
		tmp = fma(Float64(z * b), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.65e-71], N[(N[(a * b + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 2.1e-21], N[(a * t + x), $MachinePrecision], N[(N[(z * b), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6500000000000001e-71

   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 50.4%

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.4%

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

      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. lower-fma.f64 50.4%

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

   6. Applied rewrites 50.4%

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

   if -1.6500000000000001e-71 < z < 2.1000000000000001e-21

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

   if 2.1000000000000001e-21 < 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 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 70.3%

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

   4. Applied rewrites 70.3%

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

   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 50.7%

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

   7. Applied rewrites 50.7%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z\right)} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.7%

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

   9. Applied rewrites 50.7%

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

Alternative 6: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(z \cdot b, a, x\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{-50}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* z b) a x)))
   (if (<= z -6.6e-50)
     (+ x (* y z))
     (if (<= z -5.5e-74) t_1 (if (<= z 2.1e-21) (fma a t x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z * b), a, x);
	double tmp;
	if (z <= -6.6e-50) {
		tmp = x + (y * z);
	} else if (z <= -5.5e-74) {
		tmp = t_1;
	} else if (z <= 2.1e-21) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(z * b), a, x)
	tmp = 0.0
	if (z <= -6.6e-50)
		tmp = Float64(x + Float64(y * z));
	elseif (z <= -5.5e-74)
		tmp = t_1;
	elseif (z <= 2.1e-21)
		tmp = fma(a, t, 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), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -6.6e-50], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5.5e-74], t$95$1, If[LessEqual[z, 2.1e-21], N[(a * t + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5999999999999997e-50

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.6%

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

   9. Applied rewrites 50.6%

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

   if -6.5999999999999997e-50 < z < -5.5000000000000001e-74 or 2.1000000000000001e-21 < 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 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 70.3%

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

   4. Applied rewrites 70.3%

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

   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 50.7%

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

   7. Applied rewrites 50.7%

      \[\leadsto x + \color{blue}{a \cdot \left(b \cdot z\right)} \]
   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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.7%

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

   9. Applied rewrites 50.7%

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

   if -5.5000000000000001e-74 < z < 2.1000000000000001e-21

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

Alternative 7: 61.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -7.7 \cdot 10^{-50}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.7e-50)
   (+ x (* y z))
   (if (<= z 4e+196) (fma a t x) (* z (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.7e-50) {
		tmp = x + (y * z);
	} else if (z <= 4e+196) {
		tmp = fma(a, t, x);
	} else {
		tmp = z * (a * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.7e-50)
		tmp = Float64(x + Float64(y * z));
	elseif (z <= 4e+196)
		tmp = fma(a, t, x);
	else
		tmp = Float64(z * Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.7e-50], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+196], N[(a * t + x), $MachinePrecision], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.6999999999999996e-50

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.6%

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

   9. Applied rewrites 50.6%

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

   if -7.6999999999999996e-50 < z < 3.9999999999999998e196

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

   if 3.9999999999999998e196 < 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 50.4%

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 27.6%

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

   7. Applied rewrites 27.6%

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

Alternative 8: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -7.7 \cdot 10^{-50}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.7e-50)
   (+ x (* y z))
   (if (<= z 4e+196) (fma a t x) (* (* a z) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.7e-50) {
		tmp = x + (y * z);
	} else if (z <= 4e+196) {
		tmp = fma(a, t, x);
	} else {
		tmp = (a * z) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.7e-50)
		tmp = Float64(x + Float64(y * z));
	elseif (z <= 4e+196)
		tmp = fma(a, t, x);
	else
		tmp = Float64(Float64(a * z) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.7e-50], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+196], N[(a * t + x), $MachinePrecision], N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.6999999999999996e-50

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.6%

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

   9. Applied rewrites 50.6%

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

   if -7.6999999999999996e-50 < z < 3.9999999999999998e196

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

   if 3.9999999999999998e196 < 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 50.4%

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 27.6%

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

   7. Applied rewrites 27.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 28.3%

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

   9. Applied rewrites 28.3%

      \[\leadsto \left(a \cdot z\right) \cdot b \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -7.7 \cdot 10^{-50}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot z\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.7e-50)
   (+ x (* y z))
   (if (<= z 4e+196) (fma a t x) (* a (* b z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.7e-50) {
		tmp = x + (y * z);
	} else if (z <= 4e+196) {
		tmp = fma(a, t, x);
	} else {
		tmp = a * (b * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.7e-50)
		tmp = Float64(x + Float64(y * z));
	elseif (z <= 4e+196)
		tmp = fma(a, t, x);
	else
		tmp = Float64(a * Float64(b * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.7e-50], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+196], N[(a * t + x), $MachinePrecision], N[(a * N[(b * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.6999999999999996e-50

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.6%

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

   9. Applied rewrites 50.6%

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

   if -7.6999999999999996e-50 < z < 3.9999999999999998e196

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

   if 3.9999999999999998e196 < 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 50.4%

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 27.6%

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

   7. Applied rewrites 27.6%

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

Alternative 10: 59.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := x + y \cdot z\\ \mathbf{if}\;z \leq -7.7 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* y z))))
   (if (<= z -7.7e-50) t_1 (if (<= z 2.1e-18) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (y * z);
	double tmp;
	if (z <= -7.7e-50) {
		tmp = t_1;
	} else if (z <= 2.1e-18) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(y * z))
	tmp = 0.0
	if (z <= -7.7e-50)
		tmp = t_1;
	elseif (z <= 2.1e-18)
		tmp = fma(a, t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.7e-50], t$95$1, If[LessEqual[z, 2.1e-18], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6999999999999996e-50 or 2.1e-18 < 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. 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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 50.6%

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

   9. Applied rewrites 50.6%

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

   if -7.6999999999999996e-50 < z < 2.1e-18

   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

Alternative 11: 55.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+133}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.6e+133) (* y z) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.6e+133) {
		tmp = y * z;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.6e+133)
		tmp = Float64(y * z);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.6e+133], N[(y * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5999999999999998e133

   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 50.4%

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.4%

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

      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. lower-fma.f64 50.4%

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

   6. Applied rewrites 50.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot z \]
   8. Step-by-step derivation

      1. lower-*.f64 27.6%

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in y around inf

       \[\leadsto y \cdot z \]
   11. Step-by-step derivation

   12. Applied rewrites 27.0%

       \[\leadsto y \cdot z \]

   if -3.5999999999999998e133 < 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. 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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. remove-double-neg N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 94.8%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. add-flip N/A

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

      21. sub-flip N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. remove-double-neg N/A

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

      25. lower-fma.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   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 52.2%

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

   6. Applied rewrites 52.2%

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

      1. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      2. +-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      3. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      4. *-commutative 52.2%

         \[\leadsto x + a \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto x + a \cdot t \]

      6. distribute-rgt-in N/A

         \[\leadsto x + a \cdot t \]

      7. *-commutative N/A

         \[\leadsto x + a \cdot t \]

      8. lift-*.f64 52.2%

         \[\leadsto x + a \cdot t \]

      9. associate-*l* 52.2%

         \[\leadsto x + a \cdot t \]

      10. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      11. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      12. associate-+r+ N/A

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

      13. +-commutative N/A

          \[\leadsto x + a \cdot t \]

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

          \[\leadsto x + a \cdot t \]

      16. lift-*.f64 52.2%

          \[\leadsto x + a \cdot t \]

      17. lift-+.f64 N/A

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

   8. Applied rewrites 52.2%

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

Alternative 12: 39.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -7.7 \cdot 10^{-50}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.7e-50) (* y z) (if (<= z 2.3e-10) (* a t) (* y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.7e-50) {
		tmp = y * z;
	} else if (z <= 2.3e-10) {
		tmp = a * t;
	} else {
		tmp = y * z;
	}
	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 (z <= (-7.7d-50)) then
        tmp = y * z
    else if (z <= 2.3d-10) then
        tmp = a * t
    else
        tmp = y * z
    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 (z <= -7.7e-50) {
		tmp = y * z;
	} else if (z <= 2.3e-10) {
		tmp = a * t;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -7.7e-50:
		tmp = y * z
	elif z <= 2.3e-10:
		tmp = a * t
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.7e-50)
		tmp = Float64(y * z);
	elseif (z <= 2.3e-10)
		tmp = Float64(a * t);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -7.7e-50)
		tmp = y * z;
	elseif (z <= 2.3e-10)
		tmp = a * t;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.7e-50], N[(y * z), $MachinePrecision], If[LessEqual[z, 2.3e-10], N[(a * t), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6999999999999996e-50 or 2.3000000000000001e-10 < 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 50.4%

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.4%

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

      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. lower-fma.f64 50.4%

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

   6. Applied rewrites 50.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot z \]
   8. Step-by-step derivation

      1. lower-*.f64 27.6%

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in y around inf

       \[\leadsto y \cdot z \]
   11. Step-by-step derivation

   12. Applied rewrites 27.0%

       \[\leadsto y \cdot z \]

   if -7.6999999999999996e-50 < z < 2.3000000000000001e-10

   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 51.9%

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

   4. Applied rewrites 51.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 28.5%

         \[\leadsto a \cdot t \]

   7. Applied rewrites 28.5%

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

Alternative 13: 28.5% accurate, 5.3× speedup

Math

\[a \cdot t \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 = a * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * t), $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. 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 51.9%

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

4. Applied rewrites 51.9%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 28.5%

      \[\leadsto a \cdot t \]

7. Applied rewrites 28.5%

   \[\leadsto a \cdot \color{blue}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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)))