Linear.Quaternion:$c/ from linear-1.19.1.3, A

Percentage Accurate: 98.2% → 98.9%

Time: 4.4s

Alternatives: 11

Speedup: 1.8×

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z * N[(z + z), $MachinePrecision] + N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 99.2

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.2

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) + z\\ \mathbf{elif}\;t\_0 \leq 10^{+128}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z))))
   (if (<= t_0 -2e-218)
     (+ (fma y x z) z)
     (if (<= t_0 1e+128) (* (* 3.0 z) z) (fma z z (fma x y 2.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * y) + (z * z)) + (z * z)) + (z * z);
	double tmp;
	if (t_0 <= -2e-218) {
		tmp = fma(y, x, z) + z;
	} else if (t_0 <= 1e+128) {
		tmp = (3.0 * z) * z;
	} else {
		tmp = fma(z, z, fma(x, y, 2.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * y) + Float64(z * z)) + Float64(z * z)) + Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2e-218)
		tmp = Float64(fma(y, x, z) + z);
	elseif (t_0 <= 1e+128)
		tmp = Float64(Float64(3.0 * z) * z);
	else
		tmp = fma(z, z, fma(x, y, 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-218], N[(N[(y * x + z), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$0, 1e+128], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision], N[(z * z + N[(x * y + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z z)) (*.f64 z z)) (*.f64 z z)) < -2.0000000000000001e-218

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 100.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot z + x \cdot y} \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, x \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

      3. lower-*.f64 93.1

         \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

   9. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, y \cdot x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.1%

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

   if -2.0000000000000001e-218 < (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z z)) (*.f64 z z)) (*.f64 z z)) < 1.0000000000000001e128

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      5. lower-*.f64 60.2

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 60.2%

      \[\leadsto \left(3 \cdot z\right) \cdot \color{blue}{z} \]

   if 1.0000000000000001e128 < (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z z)) (*.f64 z z)) (*.f64 z z))

   1. Initial program 98.3%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 98.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-+l+ N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. distribute-lft-in N/A

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

   6. Applied rewrites 87.0%

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

Alternative 3: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-67} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-140}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 2 + z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -4e-67) (not (<= (* x y) 5e-140)))
   (fma z (+ 2.0 z) (* x y))
   (fma z (+ z z) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -4e-67) || !((x * y) <= 5e-140)) {
		tmp = fma(z, (2.0 + z), (x * y));
	} else {
		tmp = fma(z, (z + z), (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -4e-67) || !(Float64(x * y) <= 5e-140))
		tmp = fma(z, Float64(2.0 + z), Float64(x * y));
	else
		tmp = fma(z, Float64(z + z), Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e-67], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-140]], $MachinePrecision]], N[(z * N[(2.0 + z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999977e-67 or 5.00000000000000015e-140 < (*.f64 x y)

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 98.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

   if -3.99999999999999977e-67 < (*.f64 x y) < 5.00000000000000015e-140

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{{z}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 79.6

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

   7. Applied rewrites 79.6%

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{z \cdot z}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-67} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-140}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 2 + z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-67} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-140}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 2 + z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -4e-67) (not (<= (* x y) 5e-140)))
   (fma z (+ 2.0 z) (* x y))
   (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -4e-67) || !((x * y) <= 5e-140)) {
		tmp = fma(z, (2.0 + z), (x * y));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -4e-67) || !(Float64(x * y) <= 5e-140))
		tmp = fma(z, Float64(2.0 + z), Float64(x * y));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e-67], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-140]], $MachinePrecision]], N[(z * N[(2.0 + z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999977e-67 or 5.00000000000000015e-140 < (*.f64 x y)

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 98.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

   if -3.99999999999999977e-67 < (*.f64 x y) < 5.00000000000000015e-140

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      5. lower-*.f64 79.5

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 79.6%

      \[\leadsto \left(3 \cdot z\right) \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{-67} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-140}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 2 + z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) + z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.2e-24) (+ (fma y x z) z) (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.2e-24) {
		tmp = fma(y, x, z) + z;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.2e-24)
		tmp = Float64(fma(y, x, z) + z);
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.2e-24], N[(N[(y * x + z), $MachinePrecision] + z), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1999999999999999e-24

   1. Initial program 99.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 99.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot z + x \cdot y} \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, x \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

      3. lower-*.f64 54.4

         \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

   9. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, y \cdot x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.4%

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

   if 1.1999999999999999e-24 < z

   1. Initial program 98.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      5. lower-*.f64 76.8

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   5. Applied rewrites 76.8%

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

Alternative 6: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3 \cdot z, z, x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* 3.0 z) z (* x y)))

C

double code(double x, double y, double z) {
	return fma((3.0 * z), z, (x * y));
}

Julia

function code(x, y, z)
	return fma(Float64(3.0 * z), z, Float64(x * y))
end

Wolfram

code[x_, y_, z_] := N[(N[(3.0 * z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

   3. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

   4. metadata-eval N/A

      \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, {z}^{2}, x \cdot y\right)} \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(3, \color{blue}{z \cdot z}, x \cdot y\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(3, \color{blue}{z \cdot z}, x \cdot y\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, \color{blue}{y \cdot x}\right) \]

   9. lower-*.f64 99.1

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, \color{blue}{y \cdot x}\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, y \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(3 \cdot z, \color{blue}{z}, x \cdot y\right) \]
8. Add Preprocessing

Alternative 7: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 3.0 (* z z) (* y x)))

C

double code(double x, double y, double z) {
	return fma(3.0, (z * z), (y * x));
}

Julia

function code(x, y, z)
	return fma(3.0, Float64(z * z), Float64(y * x))
end

Wolfram

code[x_, y_, z_] := N[(3.0 * N[(z * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

   3. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

   4. metadata-eval N/A

      \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, {z}^{2}, x \cdot y\right)} \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(3, \color{blue}{z \cdot z}, x \cdot y\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(3, \color{blue}{z \cdot z}, x \cdot y\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, \color{blue}{y \cdot x}\right) \]

   9. lower-*.f64 99.1

      \[\leadsto \mathsf{fma}\left(3, z \cdot z, \color{blue}{y \cdot x}\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, y \cdot x\right)} \]
6. Add Preprocessing

Alternative 8: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right) + z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 9.5e+153) (+ (fma y x z) z) (fma z z 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 9.5e+153) {
		tmp = fma(y, x, z) + z;
	} else {
		tmp = fma(z, z, 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 9.5e+153)
		tmp = Float64(fma(y, x, z) + z);
	else
		tmp = fma(z, z, 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 9.5e+153], N[(N[(y * x + z), $MachinePrecision] + z), $MachinePrecision], N[(z * z + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.4999999999999995e153

   1. Initial program 99.5%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 99.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot z + x \cdot y} \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, x \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

      3. lower-*.f64 53.3

         \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

   9. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, y \cdot x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.3%

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

   if 9.4999999999999995e153 < z

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 96.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 96.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-+l+ N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. distribute-lft-in N/A

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

   6. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 9: 29.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(z - -2\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (- z -2.0) z))

C

double code(double x, double y, double z) {
	return (z - -2.0) * z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z - (-2.0d0)) * z
end function

Java

public static double code(double x, double y, double z) {
	return (z - -2.0) * z;
}

Python

def code(x, y, z):
	return (z - -2.0) * z

Julia

function code(x, y, z)
	return Float64(Float64(z - -2.0) * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (z - -2.0) * z;
end

Wolfram

code[x_, y_, z_] := N[(N[(z - -2.0), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 99.2

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.2

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-fma.f64 N/A

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

6. Applied rewrites 73.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z \cdot \left(2 + z\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(2 + z\right) \cdot z} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 + z\right) \cdot z} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(z + 2\right)} \cdot z \]

   4. metadata-eval N/A

      \[\leadsto \left(z + \color{blue}{2 \cdot 1}\right) \cdot z \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(z - \left(\mathsf{neg}\left(2\right)\right) \cdot 1\right)} \cdot z \]

   6. *-rgt-identity N/A

      \[\leadsto \left(z - \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right) \cdot z \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(z - \left(\mathsf{neg}\left(2\right)\right)\right)} \cdot z \]

   8. metadata-eval 32.1

      \[\leadsto \left(z - \color{blue}{-2}\right) \cdot z \]

9. Applied rewrites 32.1%

   \[\leadsto \color{blue}{\left(z - -2\right) \cdot z} \]
10. Add Preprocessing

Alternative 10: 29.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, z, 2\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z z 2.0))

C

double code(double x, double y, double z) {
	return fma(z, z, 2.0);
}

Julia

function code(x, y, z)
	return fma(z, z, 2.0)
end

Wolfram

code[x_, y_, z_] := N[(z * z + 2.0), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 99.2

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.2

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+r+ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-+l+ N/A

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

   20. lift-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. distribute-lft-in N/A

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

6. Applied rewrites 64.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2\right)\right)} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{2}\right) \]
8. Step-by-step derivation

9. Applied rewrites 32.2%

   \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{2}\right) \]
10. Add Preprocessing

Alternative 11: 3.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 2.0 z))

C

double code(double x, double y, double z) {
	return 2.0 * z;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * z
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * z;
}

Python

def code(x, y, z):
	return 2.0 * z

Julia

function code(x, y, z)
	return Float64(2.0 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * z;
end

Wolfram

code[x_, y_, z_] := N[(2.0 * z), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 99.2

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.2

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-fma.f64 N/A

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

6. Applied rewrites 73.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, 2 + z, x \cdot y\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot z + x \cdot y} \]
8. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, x \cdot y\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

   3. lower-*.f64 48.9

      \[\leadsto \mathsf{fma}\left(2, z, \color{blue}{y \cdot x}\right) \]

9. Applied rewrites 48.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, y \cdot x\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto 2 \cdot \color{blue}{z} \]
11. Step-by-step derivation

12. Applied rewrites 3.5%

    \[\leadsto 2 \cdot \color{blue}{z} \]
13. Add Preprocessing

Developer Target 1: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z + y \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (* 3.0 z) z) (* y x)))

C

double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((3.0d0 * z) * z) + (y * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

Python

def code(x, y, z):
	return ((3.0 * z) * z) + (y * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(3.0 * z) * z) + Float64(y * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((3.0 * z) * z) + (y * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))