Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.6% → 95.9%

Time: 4.2s

Alternatives: 13

Speedup: 1.2×

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t + b \cdot z, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1e+179) (fma z y (fma t a x)) (fma a (+ t (* b z)) (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1e+179) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = fma(a, (t + (b * z)), fma(z, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1e+179)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = fma(a, Float64(t + Float64(b * z)), fma(z, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1e+179], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(a * N[(t + N[(b * z), $MachinePrecision]), $MachinePrecision] + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.9999999999999998e178

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 77.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -9.9999999999999998e178 < t

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 94.8

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

   3. Applied rewrites 94.8%

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

Alternative 2: 94.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. distribute-lft-out N/A

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

   17. lower-*.f64 N/A

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

   18. lower-+.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 95.9

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

3. Applied rewrites 95.9%

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

Alternative 3: 87.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\left(x + a \cdot t\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2e+95)
   (+ (+ x (* a t)) (* (* a z) b))
   (if (<= b 1.25e+117) (fma z y (fma t a x)) (fma (fma b z t) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2e+95) {
		tmp = (x + (a * t)) + ((a * z) * b);
	} else if (b <= 1.25e+117) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2e+95)
		tmp = Float64(Float64(x + Float64(a * t)) + Float64(Float64(a * z) * b));
	elseif (b <= 1.25e+117)
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2e+95], N[(N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e+117], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000004e95

   1. Initial program 92.6%

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

   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 72.5

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

   4. Applied rewrites 72.5%

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

   if -2.00000000000000004e95 < b < 1.24999999999999996e117

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 77.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if 1.24999999999999996e117 < b

   1. Initial program 92.6%

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

   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 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. add-flip N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 75.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 75.2

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

   6. Applied rewrites 75.2%

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

Alternative 4: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z y (fma t a x))))
   (if (<= y -4.4e+35) t_1 (if (<= y 2.4e-35) (fma (fma b z t) a x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, y, fma(t, a, x));
	double tmp;
	if (y <= -4.4e+35) {
		tmp = t_1;
	} else if (y <= 2.4e-35) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, y, fma(t, a, x))
	tmp = 0.0
	if (y <= -4.4e+35)
		tmp = t_1;
	elseif (y <= 2.4e-35)
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.4e+35], t$95$1, If[LessEqual[y, 2.4e-35], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.3999999999999997e35 or 2.4000000000000001e-35 < y

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 77.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 77.4

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

   8. Applied rewrites 77.4%

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

   if -4.3999999999999997e35 < y < 2.4000000000000001e-35

   1. Initial program 92.6%

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

   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 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. add-flip N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 75.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 75.2

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

   6. Applied rewrites 75.2%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, y\right) \cdot z\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma a b y) z)))
   (if (<= z -1.25e+106)
     t_1
     (if (<= z 3.3e-16)
       (fma (fma b z t) a x)
       (if (<= z 2.55e+122) (fma y z x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, y) * z;
	double tmp;
	if (z <= -1.25e+106) {
		tmp = t_1;
	} else if (z <= 3.3e-16) {
		tmp = fma(fma(b, z, t), a, x);
	} else if (z <= 2.55e+122) {
		tmp = fma(y, z, 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 <= -1.25e+106)
		tmp = t_1;
	elseif (z <= 3.3e-16)
		tmp = fma(fma(b, z, t), a, x);
	elseif (z <= 2.55e+122)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.25e+106], t$95$1, If[LessEqual[z, 3.3e-16], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 2.55e+122], N[(y * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e106 or 2.55e122 < z

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.1

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

      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 50.1

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

   6. Applied rewrites 50.1%

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

   if -1.25e106 < z < 3.29999999999999988e-16

   1. Initial program 92.6%

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

   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 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. add-flip N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 75.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 75.2

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

   6. Applied rewrites 75.2%

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

   if 3.29999999999999988e-16 < z < 2.55e122

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

Alternative 6: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-62}:\\ \;\;\;\;z \cdot \left(y + a \cdot b\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-77}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -8e-62)
   (* z (+ y (* a b)))
   (if (<= z 6e-77)
     (+ x (* a t))
     (if (<= z 2.55e+122) (fma y z x) (* (fma a b y) z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -8e-62) {
		tmp = z * (y + (a * b));
	} else if (z <= 6e-77) {
		tmp = x + (a * t);
	} else if (z <= 2.55e+122) {
		tmp = fma(y, z, x);
	} else {
		tmp = fma(a, b, y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -8e-62)
		tmp = Float64(z * Float64(y + Float64(a * b)));
	elseif (z <= 6e-77)
		tmp = Float64(x + Float64(a * t));
	elseif (z <= 2.55e+122)
		tmp = fma(y, z, x);
	else
		tmp = Float64(fma(a, b, y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -8e-62], N[(z * N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-77], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.55e+122], N[(y * z + x), $MachinePrecision], N[(N[(a * b + y), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.0000000000000003e-62

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

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

   if -8.0000000000000003e-62 < z < 6.00000000000000033e-77

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

      \[\leadsto a \cdot t \]

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.9

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

   10. Applied rewrites 52.9%

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

   if 6.00000000000000033e-77 < z < 2.55e122

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

   if 2.55e122 < z

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.1

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

      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 50.1

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

   6. Applied rewrites 50.1%

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

Alternative 7: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, y\right) \cdot z\\ \mathbf{if}\;z \leq -8 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-77}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma a b y) z)))
   (if (<= z -8e-62)
     t_1
     (if (<= z 6e-77) (+ x (* a t)) (if (<= z 2.55e+122) (fma y z x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, y) * z;
	double tmp;
	if (z <= -8e-62) {
		tmp = t_1;
	} else if (z <= 6e-77) {
		tmp = x + (a * t);
	} else if (z <= 2.55e+122) {
		tmp = fma(y, z, 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 <= -8e-62)
		tmp = t_1;
	elseif (z <= 6e-77)
		tmp = Float64(x + Float64(a * t));
	elseif (z <= 2.55e+122)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -8e-62], t$95$1, If[LessEqual[z, 6e-77], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.55e+122], N[(y * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000003e-62 or 2.55e122 < z

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.1

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

      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 50.1

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

   6. Applied rewrites 50.1%

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

   if -8.0000000000000003e-62 < z < 6.00000000000000033e-77

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

      \[\leadsto a \cdot t \]

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.9

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

   10. Applied rewrites 52.9%

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

   if 6.00000000000000033e-77 < z < 2.55e122

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

Alternative 8: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+271}:\\ \;\;\;\;z \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-77}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6.3e+271)
   (* z (* a b))
   (if (<= z -8.5e+105)
     (fma y z x)
     (if (<= z 6e-77) (+ x (* a t)) (fma y z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -6.3e+271) {
		tmp = z * (a * b);
	} else if (z <= -8.5e+105) {
		tmp = fma(y, z, x);
	} else if (z <= 6e-77) {
		tmp = x + (a * t);
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -6.3e+271)
		tmp = Float64(z * Float64(a * b));
	elseif (z <= -8.5e+105)
		tmp = fma(y, z, x);
	elseif (z <= 6e-77)
		tmp = Float64(x + Float64(a * t));
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6.3e+271], N[(z * N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -8.5e+105], N[(y * z + x), $MachinePrecision], If[LessEqual[z, 6e-77], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], N[(y * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2999999999999999e271

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

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

   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 -6.2999999999999999e271 < z < -8.49999999999999986e105 or 6.00000000000000033e-77 < z

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

   if -8.49999999999999986e105 < z < 6.00000000000000033e-77

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

      \[\leadsto a \cdot t \]

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.9

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

   10. Applied rewrites 52.9%

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

Alternative 9: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+272}:\\ \;\;\;\;a \cdot \left(b \cdot z\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-77}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.5e+272)
   (* a (* b z))
   (if (<= z -8.5e+105)
     (fma y z x)
     (if (<= z 6e-77) (+ x (* a t)) (fma y z x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.5e+272) {
		tmp = a * (b * z);
	} else if (z <= -8.5e+105) {
		tmp = fma(y, z, x);
	} else if (z <= 6e-77) {
		tmp = x + (a * t);
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.5e+272)
		tmp = Float64(a * Float64(b * z));
	elseif (z <= -8.5e+105)
		tmp = fma(y, z, x);
	elseif (z <= 6e-77)
		tmp = Float64(x + Float64(a * t));
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.5e+272], N[(a * N[(b * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -8.5e+105], N[(y * z + x), $MachinePrecision], If[LessEqual[z, 6e-77], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], N[(y * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.50000000000000023e272

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{z \cdot \left(y + a \cdot b\right)} \]
   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 50.1

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

      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 50.1

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

   6. Applied rewrites 50.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 27.0

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

   9. Applied rewrites 27.0%

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

   if -3.50000000000000023e272 < z < -8.49999999999999986e105 or 6.00000000000000033e-77 < z

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

   if -8.49999999999999986e105 < z < 6.00000000000000033e-77

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

      \[\leadsto a \cdot t \]

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.9

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

   10. Applied rewrites 52.9%

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

Alternative 10: 63.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-77}:\\ \;\;\;\;x + a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -8.5e+105) (fma y z x) (if (<= z 6e-77) (+ x (* a t)) (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -8.5e+105) {
		tmp = fma(y, z, x);
	} else if (z <= 6e-77) {
		tmp = x + (a * t);
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -8.5e+105)
		tmp = fma(y, z, x);
	elseif (z <= 6e-77)
		tmp = Float64(x + Float64(a * t));
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -8.5e+105], N[(y * z + x), $MachinePrecision], If[LessEqual[z, 6e-77], N[(x + N[(a * t), $MachinePrecision]), $MachinePrecision], N[(y * z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.49999999999999986e105 or 6.00000000000000033e-77 < z

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

   if -8.49999999999999986e105 < z < 6.00000000000000033e-77

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

      \[\leadsto a \cdot t \]

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.9

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

   10. Applied rewrites 52.9%

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

Alternative 11: 58.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+107}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -5.5e+107) (* a t) (if (<= a 1.8e+178) (fma y z x) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -5.5e+107) {
		tmp = a * t;
	} else if (a <= 1.8e+178) {
		tmp = fma(y, z, x);
	} else {
		tmp = a * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -5.5e+107)
		tmp = Float64(a * t);
	elseif (a <= 1.8e+178)
		tmp = fma(y, z, x);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -5.5e+107], N[(a * t), $MachinePrecision], If[LessEqual[a, 1.8e+178], N[(y * z + x), $MachinePrecision], N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5000000000000003e107 or 1.7999999999999999e178 < a

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

      \[\leadsto a \cdot t \]

   if -5.5000000000000003e107 < a < 1.7999999999999999e178

   1. Initial program 92.6%

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. distribute-lft-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.9

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

   3. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

         \[\leadsto \color{blue}{\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}\right) \cdot x} + z \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{a \cdot \left(t + b \cdot z\right)}{x}, x, z \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot \left(t + b \cdot z\right)}{x} + 1}, x, z \cdot y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot \left(t + b \cdot z\right)}}{x} + 1, x, z \cdot y\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(t + b \cdot z\right) \cdot a}}{x} + 1, x, z \cdot y\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t + b \cdot z\right) \cdot \frac{a}{x}} + 1, x, z \cdot y\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t + b \cdot z, \frac{a}{x}, 1\right)}, x, z \cdot y\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t + b \cdot z}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z + t}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b \cdot z} + t, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, \frac{a}{x}, 1\right), x, z \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \color{blue}{\frac{a}{x}}, 1\right), x, z \cdot y\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

      17. lower-*.f64 84.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, \color{blue}{y \cdot z}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), \frac{a}{x}, 1\right), x, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.2%

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

Alternative 12: 38.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-79}:\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -8.5e+105) (* z y) (if (<= z 6.5e-79) (* a t) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -8.5e+105) {
		tmp = z * y;
	} else if (z <= 6.5e-79) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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 <= (-8.5d+105)) then
        tmp = z * y
    else if (z <= 6.5d-79) then
        tmp = a * t
    else
        tmp = z * y
    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 <= -8.5e+105) {
		tmp = z * y;
	} else if (z <= 6.5e-79) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -8.5e+105:
		tmp = z * y
	elif z <= 6.5e-79:
		tmp = a * t
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -8.5e+105)
		tmp = Float64(z * y);
	elseif (z <= 6.5e-79)
		tmp = Float64(a * t);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -8.5e+105)
		tmp = z * y;
	elseif (z <= 6.5e-79)
		tmp = a * t;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -8.5e+105], N[(z * y), $MachinePrecision], If[LessEqual[z, 6.5e-79], N[(a * t), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.49999999999999986e105 or 6.5000000000000003e-79 < z

   1. Initial program 92.6%

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

   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 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

         \[\leadsto z \cdot \left(\left(1 + \frac{a \cdot b}{y}\right) \cdot \color{blue}{y}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(z \cdot \left(1 + \frac{a \cdot b}{y}\right)\right) \cdot \color{blue}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(1 + \frac{a \cdot b}{y}\right)\right) \cdot \color{blue}{y} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(z \cdot \left(1 + \frac{a \cdot b}{y}\right)\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(z \cdot \left(\frac{a \cdot b}{y} + 1\right)\right) \cdot y \]

      8. lift-*.f64 N/A

         \[\leadsto \left(z \cdot \left(\frac{a \cdot b}{y} + 1\right)\right) \cdot y \]

      9. associate-/l* N/A

         \[\leadsto \left(z \cdot \left(a \cdot \frac{b}{y} + 1\right)\right) \cdot y \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(z \cdot \mathsf{fma}\left(a, \frac{b}{y}, 1\right)\right) \cdot y \]

      11. lower-/.f64 46.5

          \[\leadsto \left(z \cdot \mathsf{fma}\left(a, \frac{b}{y}, 1\right)\right) \cdot y \]

   6. Applied rewrites 46.5%

      \[\leadsto \left(z \cdot \mathsf{fma}\left(a, \frac{b}{y}, 1\right)\right) \cdot \color{blue}{y} \]

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 27.6%

      \[\leadsto z \cdot y \]

   if -8.49999999999999986e105 < z < 6.5000000000000003e-79

   1. Initial program 92.6%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 28.3%

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

Alternative 13: 28.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ a \cdot t \end{array} \]

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 92.6%

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

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 50.8

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

4. Applied rewrites 50.8%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 28.3%

   \[\leadsto a \cdot t \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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)))