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

Percentage Accurate: 98.2% → 99.0%

Time: 2.9s

Alternatives: 5

Speedup: N/A×

• 34.4% 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: 99.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+202)
   (* (fma 1.0 (* 3.0 (* z (/ z x))) y) x)
   (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+202) {
		tmp = fma(1.0, (3.0 * (z * (z / x))), y) * x;
	} else {
		tmp = (((x * y) + (z * z)) + (z * z)) + (z * z);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+202)
		tmp = Float64(fma(1.0, Float64(3.0 * Float64(z * Float64(z / x))), y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(x * y) + Float64(z * z)) + Float64(z * z)) + Float64(z * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -2e+202], N[(N[(1.0 * N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999998e202

   1. Initial program 95.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 2 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2}}{x}, y\right) \cdot x \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(1, \left(2 + 1\right) \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

      11. lift-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

      5. lower-/.f64 100.0

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

   if -1.9999999999999998e202 < x

   1. Initial program 97.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{fma}\left(-1 \cdot z, -1 \cdot z, \mathsf{fma}\left(z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)\right) \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (fma (* -1.0 z) (* -1.0 z) (fma z z (fma z z (* y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return fma((-1.0 * z), (-1.0 * z), fma(z, z, fma(z, z, (y * x))));
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return fma(Float64(-1.0 * z), Float64(-1.0 * z), fma(z, z, fma(z, z, Float64(y * x))))
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(-1.0 * z), $MachinePrecision] * N[(-1.0 * z), $MachinePrecision] + N[(z * z + N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. +-commutative N/A

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

   10. *-rgt-identity N/A

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

   11. pow2 N/A

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

   12. sqr-neg-rev N/A

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

   13. metadata-eval N/A

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

   14. unswap-sqr N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-z\right)} \cdot 1, \left(\mathsf{neg}\left(z\right)\right) \cdot 1, \left(x \cdot y + z \cdot z\right) + z \cdot z\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot 1, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot 1}, \left(x \cdot y + z \cdot z\right) + z \cdot z\right) \]

   19. lower-neg.f64 N/A

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

   20. pow2 N/A

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

   21. +-commutative N/A

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(-1 \cdot z, -1 \cdot z, \mathsf{fma}\left(z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)\right) \]
6. Add Preprocessing

Alternative 3: 98.6% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + e^{\mathsf{fma}\left(\log z, 2, 0\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e-28)
   (* (fma 1.0 (* 3.0 (* z (/ z x))) y) x)
   (+ (+ (+ (* x y) (* z z)) (* z z)) (exp (fma (log z) 2.0 0.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-28) {
		tmp = fma(1.0, (3.0 * (z * (z / x))), y) * x;
	} else {
		tmp = (((x * y) + (z * z)) + (z * z)) + exp(fma(log(z), 2.0, 0.0));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e-28)
		tmp = Float64(fma(1.0, Float64(3.0 * Float64(z * Float64(z / x))), y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(x * y) + Float64(z * z)) + Float64(z * z)) + exp(fma(log(z), 2.0, 0.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -3.4e-28], N[(N[(1.0 * N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(N[Log[z], $MachinePrecision] * 2.0 + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4000000000000001e-28

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 2 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2}}{x}, y\right) \cdot x \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(1, \left(2 + 1\right) \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

      11. lift-*.f64 98.4

          \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

      5. lower-/.f64 99.9

         \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

   if -3.4000000000000001e-28 < x

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. *-rgt-identity N/A

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

      4. pow-to-exp N/A

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

      5. 1-exp N/A

         \[\leadsto \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + e^{\log z \cdot 2} \cdot \color{blue}{e^{0}} \]

      6. prod-exp N/A

         \[\leadsto \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + \color{blue}{e^{\log z \cdot 2 + 0}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + \color{blue}{e^{\log z \cdot 2 + 0}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + e^{\color{blue}{\mathsf{fma}\left(\log z, 2, 0\right)}} \]

      9. lower-log.f64 48.0

         \[\leadsto \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + e^{\mathsf{fma}\left(\color{blue}{\log z}, 2, 0\right)} \]

   4. Applied rewrites 48.0%

      \[\leadsto \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + \color{blue}{e^{\mathsf{fma}\left(\log z, 2, 0\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.1% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{fma}\left(3 \cdot \left(-1 \cdot z\right), -1 \cdot z, y \cdot x\right) \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (fma (* 3.0 (* -1.0 z)) (* -1.0 z) (* y x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return fma((3.0 * (-1.0 * z)), (-1.0 * z), (y * x));
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return fma(Float64(3.0 * Float64(-1.0 * z)), Float64(-1.0 * z), Float64(y * x))
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(3.0 * N[(-1.0 * z), $MachinePrecision]), $MachinePrecision] * N[(-1.0 * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. associate-+l+ N/A

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

   11. count-2-rev N/A

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

   12. pow2 N/A

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

   13. associate-+r+ N/A

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

   14. distribute-lft1-in N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. pow2 N/A

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

   18. sqr-neg-rev N/A

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

   19. associate-*r* N/A

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

   20. lower-fma.f64 N/A

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(3 \cdot \left(-1 \cdot z\right), -1 \cdot z, y \cdot x\right) \]
6. Add Preprocessing

Alternative 5: 72.9% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (fma 1.0 (* 3.0 (* z (/ z x))) y) x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return fma(1.0, (3.0 * (z * (z / x))), y) * x;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(fma(1.0, Float64(3.0 * Float64(z * Float64(z / x))), y) * x)
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 * N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, 2 \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2}}{x}, y\right) \cdot x \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{fma}\left(1, \left(2 + 1\right) \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{{z}^{2}}{x}, y\right) \cdot x \]

   10. pow2 N/A

       \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

   11. lift-*.f64 91.1

       \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

5. Applied rewrites 91.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \frac{z \cdot z}{x}, y\right) \cdot x \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

   5. lower-/.f64 92.3

      \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]

7. Applied rewrites 92.3%

   \[\leadsto \mathsf{fma}\left(1, 3 \cdot \left(z \cdot \frac{z}{x}\right), y\right) \cdot x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))