Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.4% → 94.4%

Time: 3.1s

Alternatives: 11

Speedup: 1.3×

• 68.0% 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: 92.4% 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: 94.4% 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 92.4%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 94.4%

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

   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 94.4%

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

3. Applied rewrites 94.4%

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

Alternative 2: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \left(x + a \cdot t\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (+ (+ x (* a t)) (* (* a z) b))))
  (if (<= b -3.5e+70)
    t_1
    (if (<= b 1.05e+101) (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 + (a * t)) + ((a * z) * b);
	double tmp;
	if (b <= -3.5e+70) {
		tmp = t_1;
	} else if (b <= 1.05e+101) {
		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(Float64(x + Float64(a * t)) + Float64(Float64(a * z) * b))
	tmp = 0.0
	if (b <= -3.5e+70)
		tmp = t_1;
	elseif (b <= 1.05e+101)
		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[(N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e+70], t$95$1, If[LessEqual[b, 1.05e+101], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5e70 or 1.05e101 < b

   1. Initial program 92.4%

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

   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 72.0%

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

   4. Applied rewrites 72.0%

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

   if -3.5e70 < b < 1.05e101

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 77.1%

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

Alternative 3: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := x + \mathsf{fma}\left(a, t, a \cdot \left(b \cdot z\right)\right)\\ \mathbf{if}\;b \leq -5.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (+ x (fma a t (* a (* b z))))))
  (if (<= b -5.8)
    t_1
    (if (<= b 1.05e+101) (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 <= -5.8) {
		tmp = t_1;
	} else if (b <= 1.05e+101) {
		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 <= -5.8)
		tmp = t_1;
	elseif (b <= 1.05e+101)
		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, -5.8], t$95$1, If[LessEqual[b, 1.05e+101], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.7999999999999998 or 1.05e101 < b

   1. Initial program 92.4%

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

   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 73.9%

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

   4. Applied rewrites 73.9%

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

   if -5.7999999999999998 < b < 1.05e101

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 77.1%

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

Alternative 4: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a \cdot z, b, x\right)\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (fma (* a z) b x)))
  (if (<= b -4.2e+70)
    t_1
    (if (<= b 1.5e+132) (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 = fma((a * z), b, x);
	double tmp;
	if (b <= -4.2e+70) {
		tmp = t_1;
	} else if (b <= 1.5e+132) {
		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 = fma(Float64(a * z), b, x)
	tmp = 0.0
	if (b <= -4.2e+70)
		tmp = t_1;
	elseif (b <= 1.5e+132)
		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[(N[(a * z), $MachinePrecision] * b + x), $MachinePrecision]}, If[LessEqual[b, -4.2e+70], t$95$1, If[LessEqual[b, 1.5e+132], N[(t * a + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.2000000000000002e70 or 1.4999999999999999e132 < b

   1. Initial program 92.4%

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

   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 71.5%

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

   4. Applied rewrites 71.5%

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

   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 51.3%

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

   7. Applied rewrites 51.3%

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

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 52.8%

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

   9. Applied rewrites 52.8%

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

   if -4.2000000000000002e70 < b < 1.4999999999999999e132

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 77.1%

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

Alternative 5: 72.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a, b, y\right) \cdot z\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (* (fma a b y) z)))
  (if (<= z -2.3e+79) t_1 (if (<= z 1.02e-76) (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(a, b, y) * z;
	double tmp;
	if (z <= -2.3e+79) {
		tmp = t_1;
	} else if (z <= 1.02e-76) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(a, b, y) * z)
	tmp = 0.0
	if (z <= -2.3e+79)
		tmp = t_1;
	elseif (z <= 1.02e-76)
		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[(a * b + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.3e+79], t$95$1, If[LessEqual[z, 1.02e-76], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3e79 or 1.0200000000000001e-76 < z

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.7%

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

   4. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.7%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 50.7%

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

   6. Applied rewrites 50.7%

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

   if -2.3e79 < z < 1.0200000000000001e-76

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 52.1%

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

   6. Applied rewrites 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-rgt-in 52.1%

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

      3. *-commutative 52.1%

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

      4. *-commutative 52.1%

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

      5. associate-*l* 52.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 52.1%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 52.1%

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

   8. Applied rewrites 52.1%

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

Alternative 6: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -16000:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot z, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= t -16000.0)
  (fma a t x)
  (if (<= t 6e+76) (fma (* a z) b x) (fma a t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -16000.0) {
		tmp = fma(a, t, x);
	} else if (t <= 6e+76) {
		tmp = fma((a * z), b, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -16000.0)
		tmp = fma(a, t, x);
	elseif (t <= 6e+76)
		tmp = fma(Float64(a * z), b, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -16000.0], N[(a * t + x), $MachinePrecision], If[LessEqual[t, 6e+76], N[(N[(a * z), $MachinePrecision] * b + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -16000 or 5.9999999999999996e76 < t

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 52.1%

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

   6. Applied rewrites 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-rgt-in 52.1%

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

      3. *-commutative 52.1%

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

      4. *-commutative 52.1%

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

      5. associate-*l* 52.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 52.1%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 52.1%

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

   8. Applied rewrites 52.1%

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

   if -16000 < t < 5.9999999999999996e76

   1. Initial program 92.4%

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

   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 71.5%

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

   4. Applied rewrites 71.5%

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

   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 51.3%

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

   7. Applied rewrites 51.3%

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

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 52.8%

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

   9. Applied rewrites 52.8%

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

Alternative 7: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+70}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= b -4.2e+70)
  (* z (* a b))
  (if (<= b 1.6e+162) (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 (b <= -4.2e+70) {
		tmp = z * (a * b);
	} else if (b <= 1.6e+162) {
		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 (b <= -4.2e+70)
		tmp = Float64(z * Float64(a * b));
	elseif (b <= 1.6e+162)
		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[b, -4.2e+70], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+162], N[(a * t + x), $MachinePrecision], N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2000000000000002e70

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

   if -4.2000000000000002e70 < b < 1.6000000000000001e162

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 52.1%

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

   6. Applied rewrites 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-rgt-in 52.1%

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

      3. *-commutative 52.1%

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

      4. *-commutative 52.1%

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

      5. associate-*l* 52.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 52.1%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 52.1%

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

   8. Applied rewrites 52.1%

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

   if 1.6000000000000001e162 < b

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. 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 27.1%

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

   7. Applied rewrites 27.1%

      \[\leadsto a \cdot \color{blue}{\left(b \cdot z\right)} \]
   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 28.5%

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

   9. Applied rewrites 28.5%

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

Alternative 8: 56.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (* (* a z) b)))
  (if (<= b -4.2e+70) t_1 (if (<= b 1.6e+162) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * z) * b;
	double tmp;
	if (b <= -4.2e+70) {
		tmp = t_1;
	} else if (b <= 1.6e+162) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * z) * b)
	tmp = 0.0
	if (b <= -4.2e+70)
		tmp = t_1;
	elseif (b <= 1.6e+162)
		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[(a * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -4.2e+70], t$95$1, If[LessEqual[b, 1.6e+162], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.2000000000000002e70 or 1.6000000000000001e162 < b

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. 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 27.1%

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

   7. Applied rewrites 27.1%

      \[\leadsto a \cdot \color{blue}{\left(b \cdot z\right)} \]
   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 28.5%

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

   9. Applied rewrites 28.5%

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

   if -4.2000000000000002e70 < b < 1.6000000000000001e162

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 52.1%

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

   6. Applied rewrites 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-rgt-in 52.1%

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

      3. *-commutative 52.1%

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

      4. *-commutative 52.1%

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

      5. associate-*l* 52.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 52.1%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 52.1%

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

   8. Applied rewrites 52.1%

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

Alternative 9: 55.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := a \cdot \left(b \cdot z\right)\\ \mathbf{if}\;b \leq -9.5 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (* a (* b z))))
  (if (<= b -9.5e+112) t_1 (if (<= b 1.6e+162) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (b * z);
	double tmp;
	if (b <= -9.5e+112) {
		tmp = t_1;
	} else if (b <= 1.6e+162) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(b * z))
	tmp = 0.0
	if (b <= -9.5e+112)
		tmp = t_1;
	elseif (b <= 1.6e+162)
		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[(a * N[(b * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9.5e+112], t$95$1, If[LessEqual[b, 1.6e+162], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -9.5000000000000008e112 or 1.6000000000000001e162 < b

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 50.7%

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

   4. Applied rewrites 50.7%

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

   5. 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 27.1%

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

   7. Applied rewrites 27.1%

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

   if -9.5000000000000008e112 < b < 1.6000000000000001e162

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.4%

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

      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 94.4%

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

   3. Applied rewrites 94.4%

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

   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 52.1%

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

   6. Applied rewrites 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-rgt-in 52.1%

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

      3. *-commutative 52.1%

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

      4. *-commutative 52.1%

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

      5. associate-*l* 52.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+r+ N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 52.1%

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 52.1%

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

   8. Applied rewrites 52.1%

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

Alternative 10: 52.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 94.4%

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

   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 94.4%

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

3. Applied rewrites 94.4%

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

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 52.1%

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

6. Applied rewrites 52.1%

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

   1. *-commutative 52.1%

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

   2. distribute-rgt-in 52.1%

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

   3. *-commutative 52.1%

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

   4. *-commutative 52.1%

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

   5. associate-*l* 52.1%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 52.1%

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. +-commutative N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 52.1%

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

8. Applied rewrites 52.1%

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

Alternative 11: 27.3% accurate, 5.2× 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 92.4%

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

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 50.1%

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 27.3%

      \[\leadsto a \cdot t \]

7. Applied rewrites 27.3%

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

Reproduce

herbie shell --seed 2025212 
(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)))