bug366 (missed optimization)

Percentage Accurate: 44.9% → 99.0%

Time: 3.1s

Alternatives: 9

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (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 = sqrt(((x * x) + ((y * y) + (z * z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (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 = sqrt(((x * x) + ((y * y) + (z * z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{+120}:\\ \;\;\;\;z\_m \cdot \mathsf{fma}\left(\frac{0.5}{z\_m}, \frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{z\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{z\_m}{y\_m}, \frac{z\_m}{y\_m}, \mathsf{fma}\left(x\_m, \frac{x\_m}{y\_m \cdot y\_m}, 1\right)\right)} \cdot \left|y\_m\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (if (<= y_m 1.05e+120)
   (* z_m (fma (/ 0.5 z_m) (/ (fma y_m y_m (* x_m x_m)) z_m) 1.0))
   (*
    (sqrt (fma (/ z_m y_m) (/ z_m y_m) (fma x_m (/ x_m (* y_m y_m)) 1.0)))
    (fabs y_m))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.05e+120) {
		tmp = z_m * fma((0.5 / z_m), (fma(y_m, y_m, (x_m * x_m)) / z_m), 1.0);
	} else {
		tmp = sqrt(fma((z_m / y_m), (z_m / y_m), fma(x_m, (x_m / (y_m * y_m)), 1.0))) * fabs(y_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.05e+120)
		tmp = Float64(z_m * fma(Float64(0.5 / z_m), Float64(fma(y_m, y_m, Float64(x_m * x_m)) / z_m), 1.0));
	else
		tmp = Float64(sqrt(fma(Float64(z_m / y_m), Float64(z_m / y_m), fma(x_m, Float64(x_m / Float64(y_m * y_m)), 1.0))) * abs(y_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := If[LessEqual[y$95$m, 1.05e+120], N[(z$95$m * N[(N[(0.5 / z$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(z$95$m / y$95$m), $MachinePrecision] * N[(z$95$m / y$95$m), $MachinePrecision] + N[(x$95$m * N[(x$95$m / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[y$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e120

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 + \frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 82.9

         \[\leadsto z \cdot \left(1 + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{\color{blue}{2}}}\right) \]

   4. Applied rewrites 82.9%

      \[\leadsto \color{blue}{z \cdot \left(1 + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 85.3

          \[\leadsto z \cdot \mathsf{fma}\left(\frac{0.5}{z}, \frac{{x}^{2} + {y}^{2}}{\color{blue}{z}}, 1\right) \]

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 85.3

          \[\leadsto z \cdot \mathsf{fma}\left(\frac{0.5}{z}, \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z}, 1\right) \]

   6. Applied rewrites 85.3%

      \[\leadsto z \cdot \mathsf{fma}\left(\frac{0.5}{z}, \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z}}, 1\right) \]

   if 1.05e120 < y

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 44.9

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

   3. Applied rewrites 44.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. sum-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fabs.f64 29.1

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

   5. Applied rewrites 29.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z \cdot z + x \cdot x}}{y \cdot y} + 1} \cdot \left|y\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z \cdot z} + x \cdot x}{y \cdot y} + 1} \cdot \left|y\right| \]

      7. div-add N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 57.0

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

   7. Applied rewrites 57.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{z}{y}, \frac{z}{y}, \mathsf{fma}\left(x, \frac{x}{y \cdot y}, 1\right)\right)}} \cdot \left|y\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;z\_m \cdot \mathsf{fma}\left(\frac{0.5}{z\_m}, \frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{z\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{-\log z\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (if (<= y_m 1.32e+154)
   (* z_m (fma (/ 0.5 z_m) (/ (fma y_m y_m (* x_m x_m)) z_m) 1.0))
   (/ 1.0 (exp (- (log z_m))))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.32e+154) {
		tmp = z_m * fma((0.5 / z_m), (fma(y_m, y_m, (x_m * x_m)) / z_m), 1.0);
	} else {
		tmp = 1.0 / exp(-log(z_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.32e+154)
		tmp = Float64(z_m * fma(Float64(0.5 / z_m), Float64(fma(y_m, y_m, Float64(x_m * x_m)) / z_m), 1.0));
	else
		tmp = Float64(1.0 / exp(Float64(-log(z_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := If[LessEqual[y$95$m, 1.32e+154], N[(z$95$m * N[(N[(0.5 / z$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[(-N[Log[z$95$m], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.31999999999999998e154

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 + \frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 82.9

         \[\leadsto z \cdot \left(1 + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{\color{blue}{2}}}\right) \]

   4. Applied rewrites 82.9%

      \[\leadsto \color{blue}{z \cdot \left(1 + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 85.3

          \[\leadsto z \cdot \mathsf{fma}\left(\frac{0.5}{z}, \frac{{x}^{2} + {y}^{2}}{\color{blue}{z}}, 1\right) \]

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 85.3

          \[\leadsto z \cdot \mathsf{fma}\left(\frac{0.5}{z}, \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z}, 1\right) \]

   6. Applied rewrites 85.3%

      \[\leadsto z \cdot \mathsf{fma}\left(\frac{0.5}{z}, \color{blue}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z}}, 1\right) \]

   if 1.31999999999999998e154 < y

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.8

         \[\leadsto e^{\color{blue}{\log \left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 41.8

         \[\leadsto e^{\log \color{blue}{\left(\mathsf{fma}\left(x, x, y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 41.8

          \[\leadsto e^{\log \left(\mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(z, z, y \cdot y\right)}\right)\right) \cdot 0.5} \]

   3. Applied rewrites 41.8%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(z, z, y \cdot y\right)\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. lower-log.f64 N/A

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

      3. lower-/.f64 89.3

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

   6. Applied rewrites 89.3%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log z\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto e^{\log z} \]

      7. lower-log.f64 89.3

         \[\leadsto e^{\log z} \]

   8. Applied rewrites 89.3%

      \[\leadsto e^{\log z} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log z}} \]

      2. sinh-+-cosh-rev N/A

         \[\leadsto \color{blue}{\cosh \log z + \sinh \log z} \]

      3. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\cosh \log z \cdot \cosh \log z - \sinh \log z \cdot \sinh \log z}{\cosh \log z - \sinh \log z}} \]

      4. sinh-cosh N/A

         \[\leadsto \frac{\color{blue}{1}}{\cosh \log z - \sinh \log z} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(\log z\right)}}} \]

   10. Applied rewrites 89.3%

       \[\leadsto \color{blue}{\frac{1}{e^{-\log z}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{e^{-\log z\_m}}\\ \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{-256}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{z\_m \cdot z\_m} \cdot 0.5, z\_m, z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (exp (- (log z_m))))))
   (if (<= y_m 1.4e-256)
     t_0
     (if (<= y_m 1.32e+154)
       (fma (* (/ (fma y_m y_m (* x_m x_m)) (* z_m z_m)) 0.5) z_m z_m)
       t_0))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	double t_0 = 1.0 / exp(-log(z_m));
	double tmp;
	if (y_m <= 1.4e-256) {
		tmp = t_0;
	} else if (y_m <= 1.32e+154) {
		tmp = fma(((fma(y_m, y_m, (x_m * x_m)) / (z_m * z_m)) * 0.5), z_m, z_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	t_0 = Float64(1.0 / exp(Float64(-log(z_m))))
	tmp = 0.0
	if (y_m <= 1.4e-256)
		tmp = t_0;
	elseif (y_m <= 1.32e+154)
		tmp = fma(Float64(Float64(fma(y_m, y_m, Float64(x_m * x_m)) / Float64(z_m * z_m)) * 0.5), z_m, z_m);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(1.0 / N[Exp[(-N[Log[z$95$m], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 1.4e-256], t$95$0, If[LessEqual[y$95$m, 1.32e+154], N[(N[(N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * z$95$m + z$95$m), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.40000000000000012e-256 or 1.31999999999999998e154 < y

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.8

         \[\leadsto e^{\color{blue}{\log \left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 41.8

         \[\leadsto e^{\log \color{blue}{\left(\mathsf{fma}\left(x, x, y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 41.8

          \[\leadsto e^{\log \left(\mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(z, z, y \cdot y\right)}\right)\right) \cdot 0.5} \]

   3. Applied rewrites 41.8%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(z, z, y \cdot y\right)\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. lower-log.f64 N/A

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

      3. lower-/.f64 89.3

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

   6. Applied rewrites 89.3%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log z\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto e^{\log z} \]

      7. lower-log.f64 89.3

         \[\leadsto e^{\log z} \]

   8. Applied rewrites 89.3%

      \[\leadsto e^{\log z} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log z}} \]

      2. sinh-+-cosh-rev N/A

         \[\leadsto \color{blue}{\cosh \log z + \sinh \log z} \]

      3. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\cosh \log z \cdot \cosh \log z - \sinh \log z \cdot \sinh \log z}{\cosh \log z - \sinh \log z}} \]

      4. sinh-cosh N/A

         \[\leadsto \frac{\color{blue}{1}}{\cosh \log z - \sinh \log z} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(\log z\right)}}} \]

   10. Applied rewrites 89.3%

       \[\leadsto \color{blue}{\frac{1}{e^{-\log z}}} \]

   if 1.40000000000000012e-256 < y < 1.31999999999999998e154

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 + \frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 82.9

         \[\leadsto z \cdot \left(1 + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{\color{blue}{2}}}\right) \]

   4. Applied rewrites 82.9%

      \[\leadsto \color{blue}{z \cdot \left(1 + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 82.9

         \[\leadsto \mathsf{fma}\left(0.5 \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}, \color{blue}{z}, z\right) \]

   6. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z \cdot z} \cdot 0.5, \color{blue}{z}, z\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{e^{-\log z\_m}}\\ \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z\_m \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y\_m, y\_m, \mathsf{fma}\left(z\_m, z\_m, x\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (exp (- (log z_m))))))
   (if (<= z_m 8.5e-157)
     t_0
     (if (<= z_m 1.12e+154)
       (sqrt (fma y_m y_m (fma z_m z_m (* x_m x_m))))
       t_0))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	double t_0 = 1.0 / exp(-log(z_m));
	double tmp;
	if (z_m <= 8.5e-157) {
		tmp = t_0;
	} else if (z_m <= 1.12e+154) {
		tmp = sqrt(fma(y_m, y_m, fma(z_m, z_m, (x_m * x_m))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	t_0 = Float64(1.0 / exp(Float64(-log(z_m))))
	tmp = 0.0
	if (z_m <= 8.5e-157)
		tmp = t_0;
	elseif (z_m <= 1.12e+154)
		tmp = sqrt(fma(y_m, y_m, fma(z_m, z_m, Float64(x_m * x_m))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(1.0 / N[Exp[(-N[Log[z$95$m], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z$95$m, 8.5e-157], t$95$0, If[LessEqual[z$95$m, 1.12e+154], N[Sqrt[N[(y$95$m * y$95$m + N[(z$95$m * z$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < 8.49999999999999976e-157 or 1.11999999999999994e154 < z

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.8

         \[\leadsto e^{\color{blue}{\log \left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 41.8

         \[\leadsto e^{\log \color{blue}{\left(\mathsf{fma}\left(x, x, y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 41.8

          \[\leadsto e^{\log \left(\mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(z, z, y \cdot y\right)}\right)\right) \cdot 0.5} \]

   3. Applied rewrites 41.8%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(z, z, y \cdot y\right)\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. lower-log.f64 N/A

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

      3. lower-/.f64 89.3

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

   6. Applied rewrites 89.3%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log z\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto e^{\log z} \]

      7. lower-log.f64 89.3

         \[\leadsto e^{\log z} \]

   8. Applied rewrites 89.3%

      \[\leadsto e^{\log z} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log z}} \]

      2. sinh-+-cosh-rev N/A

         \[\leadsto \color{blue}{\cosh \log z + \sinh \log z} \]

      3. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\cosh \log z \cdot \cosh \log z - \sinh \log z \cdot \sinh \log z}{\cosh \log z - \sinh \log z}} \]

      4. sinh-cosh N/A

         \[\leadsto \frac{\color{blue}{1}}{\cosh \log z - \sinh \log z} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(\log z\right)}}} \]

   10. Applied rewrites 89.3%

       \[\leadsto \color{blue}{\frac{1}{e^{-\log z}}} \]

   if 8.49999999999999976e-157 < z < 1.11999999999999994e154

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 44.9

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

   3. Applied rewrites 44.9%

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

Alternative 5: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := e^{\log z\_m}\\ \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z\_m \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y\_m, y\_m, \mathsf{fma}\left(z\_m, z\_m, x\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (exp (log z_m))))
   (if (<= z_m 8.5e-157)
     t_0
     (if (<= z_m 1.12e+154)
       (sqrt (fma y_m y_m (fma z_m z_m (* x_m x_m))))
       t_0))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	double t_0 = exp(log(z_m));
	double tmp;
	if (z_m <= 8.5e-157) {
		tmp = t_0;
	} else if (z_m <= 1.12e+154) {
		tmp = sqrt(fma(y_m, y_m, fma(z_m, z_m, (x_m * x_m))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	t_0 = exp(log(z_m))
	tmp = 0.0
	if (z_m <= 8.5e-157)
		tmp = t_0;
	elseif (z_m <= 1.12e+154)
		tmp = sqrt(fma(y_m, y_m, fma(z_m, z_m, Float64(x_m * x_m))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[Exp[N[Log[z$95$m], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z$95$m, 8.5e-157], t$95$0, If[LessEqual[z$95$m, 1.12e+154], N[Sqrt[N[(y$95$m * y$95$m + N[(z$95$m * z$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < 8.49999999999999976e-157 or 1.11999999999999994e154 < z

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 41.8

         \[\leadsto e^{\color{blue}{\log \left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 41.8

         \[\leadsto e^{\log \color{blue}{\left(\mathsf{fma}\left(x, x, y \cdot y + z \cdot z\right)\right)} \cdot 0.5} \]

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 41.8

          \[\leadsto e^{\log \left(\mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(z, z, y \cdot y\right)}\right)\right) \cdot 0.5} \]

   3. Applied rewrites 41.8%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(z, z, y \cdot y\right)\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. lower-log.f64 N/A

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

      3. lower-/.f64 89.3

         \[\leadsto e^{-1 \cdot \log \left(\frac{1}{z}\right)} \]

   6. Applied rewrites 89.3%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-1 \cdot \color{blue}{\log \left(\frac{1}{z}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)} \]

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log z\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto e^{\log z} \]

      7. lower-log.f64 89.3

         \[\leadsto e^{\log z} \]

   8. Applied rewrites 89.3%

      \[\leadsto e^{\log z} \]

   if 8.49999999999999976e-157 < z < 1.11999999999999994e154

   1. Initial program 44.9%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 44.9

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

   3. Applied rewrites 44.9%

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

Alternative 6: 44.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \sqrt{\mathsf{fma}\left(y\_m, y\_m, \mathsf{fma}\left(z\_m, z\_m, x\_m \cdot x\_m\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (sqrt (fma y_m y_m (fma z_m z_m (* x_m x_m)))))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return sqrt(fma(y_m, y_m, fma(z_m, z_m, (x_m * x_m))));
}

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return sqrt(fma(y_m, y_m, fma(z_m, z_m, Float64(x_m * x_m))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[Sqrt[N[(y$95$m * y$95$m + N[(z$95$m * z$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 44.9%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 44.9

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

3. Applied rewrites 44.9%

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

Alternative 7: 5.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \frac{y\_m}{x\_m} \cdot \left|x\_m\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (* (/ y_m x_m) (fabs x_m)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return (y_m / x_m) * fabs(x_m);
}

Fortran

x_m =     private
y_m =     private
z_m =     private
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
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_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = (y_m / x_m) * abs(x_m)
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
assert x_m < y_m && y_m < z_m;
public static double code(double x_m, double y_m, double z_m) {
	return (y_m / x_m) * Math.abs(x_m);
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return (y_m / x_m) * math.fabs(x_m)

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(Float64(y_m / x_m) * abs(x_m))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_m, y_m, z_m)
	tmp = (y_m / x_m) * abs(x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.9%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. sum-to-mult N/A

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

   4. sqrt-prod N/A

      \[\leadsto \color{blue}{\sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \sqrt{x \cdot x}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \sqrt{\color{blue}{x \cdot x}} \]

   6. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \color{blue}{\left|x\right|} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \left|x\right|} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}}} \cdot \left|x\right| \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\frac{y \cdot y + z \cdot z}{x \cdot x} + 1}} \cdot \left|x\right| \]

   10. lower-+.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{y \cdot y + z \cdot z}{x \cdot x} + 1}} \cdot \left|x\right| \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{y \cdot y + z \cdot z}{x \cdot x}} + 1} \cdot \left|x\right| \]

   12. lift-+.f64 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{y \cdot y + z \cdot z}}{x \cdot x} + 1} \cdot \left|x\right| \]

   13. +-commutative N/A

       \[\leadsto \sqrt{\frac{\color{blue}{z \cdot z + y \cdot y}}{x \cdot x} + 1} \cdot \left|x\right| \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{z \cdot z} + y \cdot y}{x \cdot x} + 1} \cdot \left|x\right| \]

   15. lower-fma.f64 N/A

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

   16. lower-fabs.f64 11.9

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

3. Applied rewrites 11.9%

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

4. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\left(y \cdot \sqrt{\frac{1}{{x}^{2}}}\right)} \cdot \left|x\right| \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(y \cdot \color{blue}{\sqrt{\frac{1}{{x}^{2}}}}\right) \cdot \left|x\right| \]

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 5.6

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

6. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\left(y \cdot \sqrt{\frac{1}{{x}^{2}}}\right)} \cdot \left|x\right| \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{y}{\color{blue}{x}} \cdot \left|x\right| \]
8. Step-by-step derivation

   1. lower-/.f64 5.7

      \[\leadsto \frac{y}{x} \cdot \left|x\right| \]

9. Applied rewrites 5.7%

   \[\leadsto \frac{y}{\color{blue}{x}} \cdot \left|x\right| \]
10. Add Preprocessing

Alternative 8: 3.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ 1 \cdot \left|x\_m\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (* 1.0 (fabs x_m)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return 1.0 * fabs(x_m);
}

Fortran

x_m =     private
y_m =     private
z_m =     private
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
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_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = 1.0d0 * abs(x_m)
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
assert x_m < y_m && y_m < z_m;
public static double code(double x_m, double y_m, double z_m) {
	return 1.0 * Math.abs(x_m);
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return 1.0 * math.fabs(x_m)

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(1.0 * abs(x_m))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_m, y_m, z_m)
	tmp = 1.0 * abs(x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(1.0 * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.9%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. sum-to-mult N/A

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

   4. sqrt-prod N/A

      \[\leadsto \color{blue}{\sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \sqrt{x \cdot x}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \sqrt{\color{blue}{x \cdot x}} \]

   6. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \color{blue}{\left|x\right|} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}} \cdot \left|x\right|} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{1 + \frac{y \cdot y + z \cdot z}{x \cdot x}}} \cdot \left|x\right| \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\frac{y \cdot y + z \cdot z}{x \cdot x} + 1}} \cdot \left|x\right| \]

   10. lower-+.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{y \cdot y + z \cdot z}{x \cdot x} + 1}} \cdot \left|x\right| \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{y \cdot y + z \cdot z}{x \cdot x}} + 1} \cdot \left|x\right| \]

   12. lift-+.f64 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{y \cdot y + z \cdot z}}{x \cdot x} + 1} \cdot \left|x\right| \]

   13. +-commutative N/A

       \[\leadsto \sqrt{\frac{\color{blue}{z \cdot z + y \cdot y}}{x \cdot x} + 1} \cdot \left|x\right| \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{z \cdot z} + y \cdot y}{x \cdot x} + 1} \cdot \left|x\right| \]

   15. lower-fma.f64 N/A

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

   16. lower-fabs.f64 11.9

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

3. Applied rewrites 11.9%

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

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \cdot \left|x\right| \]
5. Step-by-step derivation

6. Applied rewrites 3.4%

   \[\leadsto \color{blue}{1} \cdot \left|x\right| \]
7. Add Preprocessing

Alternative 9: 1.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ -x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (- x_m))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return -x_m;
}

Fortran

x_m =     private
y_m =     private
z_m =     private
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
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_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = -x_m
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
assert x_m < y_m && y_m < z_m;
public static double code(double x_m, double y_m, double z_m) {
	return -x_m;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return -x_m

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(-x_m)
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_m, y_m, z_m)
	tmp = -x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := (-x$95$m)

Derivation

1. Initial program 44.9%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

2. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot x} \]
3. Step-by-step derivation

   1. lower-*.f64 1.7

      \[\leadsto -1 \cdot \color{blue}{x} \]

4. Applied rewrites 1.7%

   \[\leadsto \color{blue}{-1 \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{x} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x\right) \]

   3. lower-neg.f64 1.7

      \[\leadsto -x \]

6. Applied rewrites 1.7%

   \[\leadsto -x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64
  (sqrt (+ (* x x) (+ (* y y) (* z z)))))