Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.6% → 98.9%

Time: 5.7s

Alternatives: 9

Speedup: 1.1×

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, c)
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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, c)
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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (fma (* z 0.0625) t (fma x y (fma (* -0.25 a) b c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma((z * 0.0625), t, fma(x, y, fma((-0.25 * a), b, c)));
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(Float64(z * 0.0625), t, fma(x, y, fma(Float64(-0.25 * a), b, c)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(z * 0.0625), $MachinePrecision] * t + N[(x * y + N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

   2. sub-flip N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   8. associate-/l* N/A

      \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

   11. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

   18. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

   19. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   24. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

3. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(a \cdot b\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+114)
   (fma (* z 0.0625) t (fma y x c))
   (if (<= (* x y) 2e+194)
     (+ (fma (* t 0.0625) z (* -0.25 (* a b))) c)
     (fma y x (fma (* t z) 0.0625 c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -2e+114) {
		tmp = fma((z * 0.0625), t, fma(y, x, c));
	} else if ((x * y) <= 2e+194) {
		tmp = fma((t * 0.0625), z, (-0.25 * (a * b))) + c;
	} else {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+114)
		tmp = fma(Float64(z * 0.0625), t, fma(y, x, c));
	elseif (Float64(x * y) <= 2e+194)
		tmp = Float64(fma(Float64(t * 0.0625), z, Float64(-0.25 * Float64(a * b))) + c);
	else
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+114], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+194], N[(N[(N[(t * 0.0625), $MachinePrecision] * z + N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e114

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      8. associate-/l* N/A

         \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      11. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      18. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

      19. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, \color{blue}{c + x \cdot y}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lower-*.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c + x \cdot \color{blue}{y}\right) \]

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \color{blue}{c + x \cdot y}\right) \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + x \cdot \color{blue}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + y \cdot \color{blue}{x}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, y \cdot x + \color{blue}{c}\right) \]

      5. lower-fma.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, \color{blue}{x}, c\right)\right) \]

   10. Applied rewrites 73.5%

       \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, \color{blue}{x}, c\right)\right) \]

   if -2e114 < (*.f64 x y) < 1.99999999999999989e194

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      8. associate-/l* N/A

         \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      11. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      18. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

      19. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) + c \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)}\right) + c \]

      2. lower-*.f64 74.3

         \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, -0.25 \cdot \left(a \cdot \color{blue}{b}\right)\right) + c \]

   6. Applied rewrites 74.3%

      \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{-0.25 \cdot \left(a \cdot b\right)}\right) + c \]

   if 1.99999999999999989e194 < (*.f64 x y)

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)}\right) \]

      3. lower-*.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, c + 0.0625 \cdot \left(t \cdot \color{blue}{z}\right)\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{c}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{1}{16} \cdot \left(t \cdot z\right) + c\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot \frac{1}{16} + c\right) \]

      5. lower-fma.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right)\right) \]

   8. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -1e+54)
     (fma y x (fma (* t z) 0.0625 c))
     (if (<= t_1 5e+44)
       (fma y x (fma b (* a -0.25) c))
       (fma (* z 0.0625) t (fma y x c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else if (t_1 <= 5e+44) {
		tmp = fma(y, x, fma(b, (a * -0.25), c));
	} else {
		tmp = fma((z * 0.0625), t, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	elseif (t_1 <= 5e+44)
		tmp = fma(y, x, fma(b, Float64(a * -0.25), c));
	else
		tmp = fma(Float64(z * 0.0625), t, fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+44], N[(y * x + N[(b * N[(a * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)}\right) \]

      3. lower-*.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, c + 0.0625 \cdot \left(t \cdot \color{blue}{z}\right)\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{c}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{1}{16} \cdot \left(t \cdot z\right) + c\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot \frac{1}{16} + c\right) \]

      5. lower-fma.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right)\right) \]

   8. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right)\right) \]

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.9999999999999996e44

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

      3. lower-*.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, x, c + -0.25 \cdot \left(a \cdot \color{blue}{b}\right)\right) \]

   6. Applied rewrites 74.8%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + -0.25 \cdot \left(a \cdot b\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + \color{blue}{c}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + c\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + c\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(\frac{-1}{4} \cdot a\right) \cdot b + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(\frac{-1}{4} \cdot a\right) \cdot b + c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, b \cdot \left(\frac{-1}{4} \cdot a\right) + c\right) \]

      8. lower-fma.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, \color{blue}{-0.25 \cdot a}, c\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, \frac{-1}{4} \cdot \color{blue}{a}, c\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot \color{blue}{\frac{-1}{4}}, c\right)\right) \]

      11. lower-*.f64 74.8

          \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot \color{blue}{-0.25}, c\right)\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot -0.25, c\right)\right)} \]

   if 4.9999999999999996e44 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      8. associate-/l* N/A

         \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      11. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      18. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

      19. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, \color{blue}{c + x \cdot y}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lower-*.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c + x \cdot \color{blue}{y}\right) \]

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \color{blue}{c + x \cdot y}\right) \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + x \cdot \color{blue}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + y \cdot \color{blue}{x}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, y \cdot x + \color{blue}{c}\right) \]

      5. lower-fma.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, \color{blue}{x}, c\right)\right) \]

   10. Applied rewrites 73.5%

       \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, \color{blue}{x}, c\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot -0.25, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+179}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma y x (fma b (* a -0.25) c))))
   (if (<= t_1 -2e+179)
     t_2
     (if (<= t_1 5e+205) (fma y x (fma (* t z) 0.0625 c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = fma(y, x, fma(b, (a * -0.25), c));
	double tmp;
	if (t_1 <= -2e+179) {
		tmp = t_2;
	} else if (t_1 <= 5e+205) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = fma(y, x, fma(b, Float64(a * -0.25), c))
	tmp = 0.0
	if (t_1 <= -2e+179)
		tmp = t_2;
	elseif (t_1 <= 5e+205)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(y * x + N[(b * N[(a * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+179], t$95$2, If[LessEqual[t$95$1, 5e+205], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.99999999999999996e179 or 5.0000000000000002e205 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

      3. lower-*.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, x, c + -0.25 \cdot \left(a \cdot \color{blue}{b}\right)\right) \]

   6. Applied rewrites 74.8%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + -0.25 \cdot \left(a \cdot b\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + \color{blue}{c}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + c\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + c\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(\frac{-1}{4} \cdot a\right) \cdot b + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(\frac{-1}{4} \cdot a\right) \cdot b + c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, b \cdot \left(\frac{-1}{4} \cdot a\right) + c\right) \]

      8. lower-fma.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, \color{blue}{-0.25 \cdot a}, c\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, \frac{-1}{4} \cdot \color{blue}{a}, c\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot \color{blue}{\frac{-1}{4}}, c\right)\right) \]

      11. lower-*.f64 74.8

          \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot \color{blue}{-0.25}, c\right)\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot -0.25, c\right)\right)} \]

   if -1.99999999999999996e179 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.0000000000000002e205

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)}\right) \]

      3. lower-*.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, c + 0.0625 \cdot \left(t \cdot \color{blue}{z}\right)\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{c}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{1}{16} \cdot \left(t \cdot z\right) + c\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot \frac{1}{16} + c\right) \]

      5. lower-fma.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right)\right) \]

   8. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+172}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* z 0.0625) t c)))
   (if (<= t_1 -2e+172)
     t_2
     (if (<= t_1 4e+111) (fma y x (fma b (* a -0.25) c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((z * 0.0625), t, c);
	double tmp;
	if (t_1 <= -2e+172) {
		tmp = t_2;
	} else if (t_1 <= 4e+111) {
		tmp = fma(y, x, fma(b, (a * -0.25), c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(z * 0.0625), t, c)
	tmp = 0.0
	if (t_1 <= -2e+172)
		tmp = t_2;
	elseif (t_1 <= 4e+111)
		tmp = fma(y, x, fma(b, Float64(a * -0.25), c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+172], t$95$2, If[LessEqual[t$95$1, 4e+111], N[(y * x + N[(b * N[(a * -0.25), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2.0000000000000002e172 or 3.99999999999999983e111 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      8. associate-/l* N/A

         \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      11. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      18. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

      19. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, \color{blue}{c + x \cdot y}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lower-*.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c + x \cdot \color{blue}{y}\right) \]

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \color{blue}{c + x \cdot y}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.3%

       \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c\right) \]

   if -2.0000000000000002e172 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 3.99999999999999983e111

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

      3. lower-*.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, x, c + -0.25 \cdot \left(a \cdot \color{blue}{b}\right)\right) \]

   6. Applied rewrites 74.8%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + -0.25 \cdot \left(a \cdot b\right)}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + \color{blue}{c}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + c\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right) + c\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(\frac{-1}{4} \cdot a\right) \cdot b + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \left(\frac{-1}{4} \cdot a\right) \cdot b + c\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, b \cdot \left(\frac{-1}{4} \cdot a\right) + c\right) \]

      8. lower-fma.f64 74.8

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, \color{blue}{-0.25 \cdot a}, c\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, \frac{-1}{4} \cdot \color{blue}{a}, c\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot \color{blue}{\frac{-1}{4}}, c\right)\right) \]

      11. lower-*.f64 74.8

          \[\leadsto \mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot \color{blue}{-0.25}, c\right)\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(b, a \cdot -0.25, c\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* z 0.0625) t c)))
   (if (<= t_1 -1e+54) t_2 (if (<= t_1 1e+88) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((z * 0.0625), t, c);
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = t_2;
	} else if (t_1 <= 1e+88) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(z * 0.0625), t, c)
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = t_2;
	elseif (t_1 <= 1e+88)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], t$95$2, If[LessEqual[t$95$1, 1e+88], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54 or 9.99999999999999959e87 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      8. associate-/l* N/A

         \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      11. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      18. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

      19. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, \color{blue}{c + x \cdot y}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lower-*.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c + x \cdot \color{blue}{y}\right) \]

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \color{blue}{c + x \cdot y}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.3%

       \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c\right) \]

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999959e87

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)}\right) \]

      3. lower-*.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, c + 0.0625 \cdot \left(t \cdot \color{blue}{z}\right)\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 48.4%

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

Alternative 7: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* 0.0625 (* t z))))
   (if (<= t_1 -1e+54) t_2 (if (<= t_1 1e+88) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = t_2;
	} else if (t_1 <= 1e+88) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = t_2;
	elseif (t_1 <= 1e+88)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], t$95$2, If[LessEqual[t$95$1, 1e+88], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54 or 9.99999999999999959e87 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      8. associate-/l* N/A

         \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

      11. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

      18. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

      19. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

      24. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, \color{blue}{c + x \cdot y}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \frac{1}{16}, t, c + \color{blue}{x \cdot y}\right) \]

      2. lower-*.f64 73.5

         \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c + x \cdot \color{blue}{y}\right) \]

   8. Applied rewrites 73.5%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, \color{blue}{c + x \cdot y}\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)} \]

       2. lower-*.f64 28.1

          \[\leadsto 0.0625 \cdot \left(t \cdot \color{blue}{z}\right) \]

   11. Applied rewrites 28.1%

       \[\leadsto \color{blue}{0.0625 \cdot \left(t \cdot z\right)} \]

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999959e87

   1. Initial program 97.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c + \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)}\right) \]

      3. lower-*.f64 73.6

         \[\leadsto \mathsf{fma}\left(y, x, c + 0.0625 \cdot \left(t \cdot \color{blue}{z}\right)\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 48.4%

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

Alternative 8: 48.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, c\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

   4. associate--l+ N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right)\right)} + c \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{x \cdot y + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot y} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot x} + \left(\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(\frac{z \cdot t}{16} - \frac{a \cdot b}{4}\right) + c}\right) \]

3. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot 0.0625, z, \left(b \cdot a\right) \cdot -0.25\right) + c\right)} \]

4. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + \frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, c + \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)}\right) \]

   3. lower-*.f64 73.6

      \[\leadsto \mathsf{fma}\left(y, x, c + 0.0625 \cdot \left(t \cdot \color{blue}{z}\right)\right) \]

6. Applied rewrites 73.6%

   \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{c + 0.0625 \cdot \left(t \cdot z\right)}\right) \]

7. Taylor expanded in z around 0

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

9. Applied rewrites 48.4%

   \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
10. Add Preprocessing

Alternative 9: 28.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

   2. sub-flip N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right)} + c \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   8. associate-/l* N/A

      \[\leadsto \left(\color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)\right) + c \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right)} + c \]

   11. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)\right) + c \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, \mathsf{neg}\left(\frac{a \cdot b}{4}\right)\right)}\right) + c \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\frac{a \cdot b}{4}}\right)\right)\right) + c \]

   18. mult-flip N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right)\right)\right) + c \]

   19. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) + c \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)\right)\right) + c \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) + c \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{-4}}\right)\right) + c \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \left(a \cdot b\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

   24. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \mathsf{fma}\left(y, x, \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{\mathsf{neg}\left(4\right)}}\right)\right) + c \]

3. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(y, x, \left(b \cdot a\right) \cdot -0.25\right)\right)} + c \]

5. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\right)} \]

6. Taylor expanded in x around inf

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

   1. lower-*.f64 28.3

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

8. Applied rewrites 28.3%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))