(pow (+ (* z0 z0) (* z1 z1)) (* 1/2 z2))

Percentage Accurate: 76.3% → 89.4%

Time: 8.8s

Alternatives: 4

Speedup: 1.0×

• 74.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (pow (+ (* z0 z0) (* z1 z1)) (* 1/2 z2)))

C

double code(double z0, double z1, double z2) {
	return pow(((z0 * z0) + (z1 * z1)), (0.5 * z2));
}

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(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = ((z0 * z0) + (z1 * z1)) ** (0.5d0 * z2)
end function

Java

public static double code(double z0, double z1, double z2) {
	return Math.pow(((z0 * z0) + (z1 * z1)), (0.5 * z2));
}

Python

def code(z0, z1, z2):
	return math.pow(((z0 * z0) + (z1 * z1)), (0.5 * z2))

Julia

function code(z0, z1, z2)
	return Float64(Float64(z0 * z0) + Float64(z1 * z1)) ^ Float64(0.5 * z2)
end

MATLAB

function tmp = code(z0, z1, z2)
	tmp = ((z0 * z0) + (z1 * z1)) ^ (0.5 * z2);
end

Wolfram

code[z0_, z1_, z2_] := N[Power[N[(N[(z0 * z0), $MachinePrecision] + N[(z1 * z1), $MachinePrecision]), $MachinePrecision], N[(1/2 * z2), $MachinePrecision]], $MachinePrecision]

Initial Program: 76.3% accurate, 1.0× speedup

Math

\[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (pow (+ (* z0 z0) (* z1 z1)) (* 1/2 z2)))

C

double code(double z0, double z1, double z2) {
	return pow(((z0 * z0) + (z1 * z1)), (0.5 * z2));
}

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(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = ((z0 * z0) + (z1 * z1)) ** (0.5d0 * z2)
end function

Java

public static double code(double z0, double z1, double z2) {
	return Math.pow(((z0 * z0) + (z1 * z1)), (0.5 * z2));
}

Python

def code(z0, z1, z2):
	return math.pow(((z0 * z0) + (z1 * z1)), (0.5 * z2))

Julia

function code(z0, z1, z2)
	return Float64(Float64(z0 * z0) + Float64(z1 * z1)) ^ Float64(0.5 * z2)
end

MATLAB

function tmp = code(z0, z1, z2)
	tmp = ((z0 * z0) + (z1 * z1)) ^ (0.5 * z2);
end

Wolfram

code[z0_, z1_, z2_] := N[Power[N[(N[(z0 * z0), $MachinePrecision] + N[(z1 * z1), $MachinePrecision]), $MachinePrecision], N[(1/2 * z2), $MachinePrecision]], $MachinePrecision]

Alternative 1: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|z0\right|\right|\\ t_1 := \left|z0\right| \cdot \left|z0\right|\\ \mathbf{if}\;\left|z0\right| \leq \frac{6338253001141147}{79228162514264337593543950336}:\\ \;\;\;\;{\left(t\_1 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left({\left(t\_1 \cdot t\_1\right)}^{\frac{-1}{4}} \cdot z1\right) \cdot z1 + t\_0\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {t\_0}^{\left(z2 \cdot \frac{1}{2}\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (let* ((t_0 (fabs (fabs z0))) (t_1 (* (fabs z0) (fabs z0))))
  (if (<= (fabs z0) 6338253001141147/79228162514264337593543950336)
    (pow (+ t_1 (* z1 z1)) (* 1/2 z2))
    (*
     (pow (+ (* (* (pow (* t_1 t_1) -1/4) z1) z1) t_0) (* z2 1/2))
     (pow t_0 (* z2 1/2))))))

C

double code(double z0, double z1, double z2) {
	double t_0 = fabs(fabs(z0));
	double t_1 = fabs(z0) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 8e-14) {
		tmp = pow((t_1 + (z1 * z1)), (0.5 * z2));
	} else {
		tmp = pow((((pow((t_1 * t_1), -0.25) * z1) * z1) + t_0), (z2 * 0.5)) * pow(t_0, (z2 * 0.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(abs(z0))
    t_1 = abs(z0) * abs(z0)
    if (abs(z0) <= 8d-14) then
        tmp = (t_1 + (z1 * z1)) ** (0.5d0 * z2)
    else
        tmp = ((((((t_1 * t_1) ** (-0.25d0)) * z1) * z1) + t_0) ** (z2 * 0.5d0)) * (t_0 ** (z2 * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1, double z2) {
	double t_0 = Math.abs(Math.abs(z0));
	double t_1 = Math.abs(z0) * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 8e-14) {
		tmp = Math.pow((t_1 + (z1 * z1)), (0.5 * z2));
	} else {
		tmp = Math.pow((((Math.pow((t_1 * t_1), -0.25) * z1) * z1) + t_0), (z2 * 0.5)) * Math.pow(t_0, (z2 * 0.5));
	}
	return tmp;
}

Python

def code(z0, z1, z2):
	t_0 = math.fabs(math.fabs(z0))
	t_1 = math.fabs(z0) * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 8e-14:
		tmp = math.pow((t_1 + (z1 * z1)), (0.5 * z2))
	else:
		tmp = math.pow((((math.pow((t_1 * t_1), -0.25) * z1) * z1) + t_0), (z2 * 0.5)) * math.pow(t_0, (z2 * 0.5))
	return tmp

Julia

function code(z0, z1, z2)
	t_0 = abs(abs(z0))
	t_1 = Float64(abs(z0) * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 8e-14)
		tmp = Float64(t_1 + Float64(z1 * z1)) ^ Float64(0.5 * z2);
	else
		tmp = Float64((Float64(Float64(Float64((Float64(t_1 * t_1) ^ -0.25) * z1) * z1) + t_0) ^ Float64(z2 * 0.5)) * (t_0 ^ Float64(z2 * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2)
	t_0 = abs(abs(z0));
	t_1 = abs(z0) * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 8e-14)
		tmp = (t_1 + (z1 * z1)) ^ (0.5 * z2);
	else
		tmp = ((((((t_1 * t_1) ^ -0.25) * z1) * z1) + t_0) ^ (z2 * 0.5)) * (t_0 ^ (z2 * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_] := Block[{t$95$0 = N[Abs[N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 6338253001141147/79228162514264337593543950336], N[Power[N[(t$95$1 + N[(z1 * z1), $MachinePrecision]), $MachinePrecision], N[(1/2 * z2), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], -1/4], $MachinePrecision] * z1), $MachinePrecision] * z1), $MachinePrecision] + t$95$0), $MachinePrecision], N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 8e-14

   1. Initial program 76.3%

      \[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

   if 8e-14 < z0

   1. Initial program 76.3%

      \[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]
   2. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto {\left(\color{blue}{{\left(z0 \cdot z0\right)}^{1}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      2. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{1 - -1}}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      4. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{1 - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      5. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{\frac{1}{2} \cdot 2} - \left(\mathsf{neg}\left(1\right)\right)}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      6. sub-to-fraction N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      7. pow-sub N/A

         \[\leadsto {\left(\color{blue}{\frac{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      8. lower-unsound-pow.f32 N/A

         \[\leadsto {\left(\frac{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      9. lower-pow.f32 N/A

         \[\leadsto {\left(\frac{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      10. pow1/2 N/A

          \[\leadsto {\left(\frac{\color{blue}{\sqrt{z0 \cdot z0}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\sqrt{\color{blue}{z0 \cdot z0}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

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

          \[\leadsto {\left(\frac{\color{blue}{\left|z0\right|}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      13. lower-unsound-/.f64 N/A

          \[\leadsto {\left(\color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      14. lower-fabs.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{\left|z0\right|}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto {\left(\frac{\left|z0\right|}{\color{blue}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      16. metadata-eval N/A

          \[\leadsto {\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      17. metadata-eval 76.3%

          \[\leadsto {\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\color{blue}{\frac{-1}{2}}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

   3. Applied rewrites 76.3%

      \[\leadsto {\left(\color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} + z1 \cdot z1\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      3. +-commutative N/A

         \[\leadsto {\color{blue}{\left(z1 \cdot z1 + \frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto {\left(z1 \cdot z1 + \color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      5. add-to-fraction N/A

         \[\leadsto {\color{blue}{\left(\frac{\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      6. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \frac{1}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      8. pow-flip N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{{\left(z0 \cdot z0\right)}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      9. metadata-eval N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot {\left(z0 \cdot z0\right)}^{\color{blue}{\frac{1}{2}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      10. pow1/2 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\sqrt{z0 \cdot z0}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \sqrt{\color{blue}{z0 \cdot z0}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

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

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\left|z0\right|}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      13. lift-fabs.f64 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\left|z0\right|}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      14. unpow-prod-down N/A

          \[\leadsto \color{blue}{{\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)} \cdot {\left(\left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

      15. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)} \cdot {\left(\left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(z1 \cdot z1\right) + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(z1 \cdot z1\right)} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(z1 \cdot z1\right)} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      3. associate-*r* N/A

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot z1\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      4. remove-double-neg N/A

         \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z1\right)\right)\right)\right)}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(z1\right)\right)\right)\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(z1\right)\right)\right)\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      7. distribute-rgt-neg-out N/A

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z1\right)\right)\right)\right)\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      8. remove-double-neg N/A

         \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{z1}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      9. lower-*.f64 63.3%

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot z1\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto {\left(\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      12. pow-neg N/A

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      13. lower-unsound-pow.f32 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      14. lower-pow.f32 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      15. pow1/2 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\sqrt{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto {\left(\left(\frac{1}{\sqrt{\color{blue}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

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

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      18. lift-fabs.f64 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      19. lower-unsound-/.f64 72.5%

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

   7. Applied rewrites 72.5%

      \[\leadsto {\left(\color{blue}{\left(\frac{1}{\left|z0\right|} \cdot z1\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto {\left(\left(\color{blue}{\frac{1}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      2. inv-pow N/A

         \[\leadsto {\left(\left(\color{blue}{{\left(\left|z0\right|\right)}^{-1}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      3. lift-fabs.f64 N/A

         \[\leadsto {\left(\left({\color{blue}{\left(\left|z0\right|\right)}}^{-1} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

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

         \[\leadsto {\left(\left({\color{blue}{\left(\sqrt{z0 \cdot z0}\right)}}^{-1} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      5. sqrt-pow2 N/A

         \[\leadsto {\left(\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\left(\frac{-1}{2}\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      6. fabs-sqr N/A

         \[\leadsto {\left(\left({\color{blue}{\left(\left|z0 \cdot z0\right|\right)}}^{\left(\frac{-1}{2}\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

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

         \[\leadsto {\left(\left({\color{blue}{\left(\sqrt{\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)}\right)}}^{\left(\frac{-1}{2}\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      8. sqrt-pow2 N/A

         \[\leadsto {\left(\left(\color{blue}{{\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      9. lower-*.f32 N/A

         \[\leadsto {\left(\left({\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      10. lower-unsound-*.f32 N/A

          \[\leadsto {\left(\left({\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto {\left(\left({\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      12. metadata-eval N/A

          \[\leadsto {\left(\left({\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\color{blue}{\frac{-1}{4}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      13. metadata-eval N/A

          \[\leadsto {\left(\left({\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      14. lower-pow.f64 N/A

          \[\leadsto {\left(\left(\color{blue}{{\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      15. lower-unsound-*.f64 N/A

          \[\leadsto {\left(\left({\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto {\left(\left({\left(\color{blue}{\left(z0 \cdot z0\right)} \cdot \left(z0 \cdot z0\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto {\left(\left({\left(\left(z0 \cdot z0\right) \cdot \color{blue}{\left(z0 \cdot z0\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      18. metadata-eval 56.9%

          \[\leadsto {\left(\left({\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\color{blue}{\frac{-1}{4}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

   9. Applied rewrites 56.9%

      \[\leadsto {\left(\left(\color{blue}{{\left(\left(z0 \cdot z0\right) \cdot \left(z0 \cdot z0\right)\right)}^{\frac{-1}{4}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|z0\right|\right|\\ t_1 := \left|z0\right| \cdot \left|z0\right|\\ \mathbf{if}\;\left|z0\right| \leq \frac{6338253001141147}{79228162514264337593543950336}:\\ \;\;\;\;{\left(t\_1 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\sqrt{\frac{1}{t\_1}} \cdot z1\right) \cdot z1 + t\_0\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {t\_0}^{\left(z2 \cdot \frac{1}{2}\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (let* ((t_0 (fabs (fabs z0))) (t_1 (* (fabs z0) (fabs z0))))
  (if (<= (fabs z0) 6338253001141147/79228162514264337593543950336)
    (pow (+ t_1 (* z1 z1)) (* 1/2 z2))
    (*
     (pow (+ (* (* (sqrt (/ 1 t_1)) z1) z1) t_0) (* z2 1/2))
     (pow t_0 (* z2 1/2))))))

C

double code(double z0, double z1, double z2) {
	double t_0 = fabs(fabs(z0));
	double t_1 = fabs(z0) * fabs(z0);
	double tmp;
	if (fabs(z0) <= 8e-14) {
		tmp = pow((t_1 + (z1 * z1)), (0.5 * z2));
	} else {
		tmp = pow((((sqrt((1.0 / t_1)) * z1) * z1) + t_0), (z2 * 0.5)) * pow(t_0, (z2 * 0.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(abs(z0))
    t_1 = abs(z0) * abs(z0)
    if (abs(z0) <= 8d-14) then
        tmp = (t_1 + (z1 * z1)) ** (0.5d0 * z2)
    else
        tmp = ((((sqrt((1.0d0 / t_1)) * z1) * z1) + t_0) ** (z2 * 0.5d0)) * (t_0 ** (z2 * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1, double z2) {
	double t_0 = Math.abs(Math.abs(z0));
	double t_1 = Math.abs(z0) * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 8e-14) {
		tmp = Math.pow((t_1 + (z1 * z1)), (0.5 * z2));
	} else {
		tmp = Math.pow((((Math.sqrt((1.0 / t_1)) * z1) * z1) + t_0), (z2 * 0.5)) * Math.pow(t_0, (z2 * 0.5));
	}
	return tmp;
}

Python

def code(z0, z1, z2):
	t_0 = math.fabs(math.fabs(z0))
	t_1 = math.fabs(z0) * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 8e-14:
		tmp = math.pow((t_1 + (z1 * z1)), (0.5 * z2))
	else:
		tmp = math.pow((((math.sqrt((1.0 / t_1)) * z1) * z1) + t_0), (z2 * 0.5)) * math.pow(t_0, (z2 * 0.5))
	return tmp

Julia

function code(z0, z1, z2)
	t_0 = abs(abs(z0))
	t_1 = Float64(abs(z0) * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 8e-14)
		tmp = Float64(t_1 + Float64(z1 * z1)) ^ Float64(0.5 * z2);
	else
		tmp = Float64((Float64(Float64(Float64(sqrt(Float64(1.0 / t_1)) * z1) * z1) + t_0) ^ Float64(z2 * 0.5)) * (t_0 ^ Float64(z2 * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2)
	t_0 = abs(abs(z0));
	t_1 = abs(z0) * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 8e-14)
		tmp = (t_1 + (z1 * z1)) ^ (0.5 * z2);
	else
		tmp = ((((sqrt((1.0 / t_1)) * z1) * z1) + t_0) ^ (z2 * 0.5)) * (t_0 ^ (z2 * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_] := Block[{t$95$0 = N[Abs[N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 6338253001141147/79228162514264337593543950336], N[Power[N[(t$95$1 + N[(z1 * z1), $MachinePrecision]), $MachinePrecision], N[(1/2 * z2), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Sqrt[N[(1 / t$95$1), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision] * z1), $MachinePrecision] + t$95$0), $MachinePrecision], N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 8e-14

   1. Initial program 76.3%

      \[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

   if 8e-14 < z0

   1. Initial program 76.3%

      \[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]
   2. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto {\left(\color{blue}{{\left(z0 \cdot z0\right)}^{1}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      2. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{1 - -1}}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      4. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{1 - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      5. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{\frac{1}{2} \cdot 2} - \left(\mathsf{neg}\left(1\right)\right)}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      6. sub-to-fraction N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      7. pow-sub N/A

         \[\leadsto {\left(\color{blue}{\frac{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      8. lower-unsound-pow.f32 N/A

         \[\leadsto {\left(\frac{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      9. lower-pow.f32 N/A

         \[\leadsto {\left(\frac{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      10. pow1/2 N/A

          \[\leadsto {\left(\frac{\color{blue}{\sqrt{z0 \cdot z0}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\sqrt{\color{blue}{z0 \cdot z0}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

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

          \[\leadsto {\left(\frac{\color{blue}{\left|z0\right|}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      13. lower-unsound-/.f64 N/A

          \[\leadsto {\left(\color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      14. lower-fabs.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{\left|z0\right|}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto {\left(\frac{\left|z0\right|}{\color{blue}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      16. metadata-eval N/A

          \[\leadsto {\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      17. metadata-eval 76.3%

          \[\leadsto {\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\color{blue}{\frac{-1}{2}}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

   3. Applied rewrites 76.3%

      \[\leadsto {\left(\color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} + z1 \cdot z1\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      3. +-commutative N/A

         \[\leadsto {\color{blue}{\left(z1 \cdot z1 + \frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto {\left(z1 \cdot z1 + \color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      5. add-to-fraction N/A

         \[\leadsto {\color{blue}{\left(\frac{\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      6. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \frac{1}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      8. pow-flip N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{{\left(z0 \cdot z0\right)}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      9. metadata-eval N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot {\left(z0 \cdot z0\right)}^{\color{blue}{\frac{1}{2}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      10. pow1/2 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\sqrt{z0 \cdot z0}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \sqrt{\color{blue}{z0 \cdot z0}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

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

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\left|z0\right|}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      13. lift-fabs.f64 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\left|z0\right|}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      14. unpow-prod-down N/A

          \[\leadsto \color{blue}{{\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)} \cdot {\left(\left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

      15. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)} \cdot {\left(\left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(z1 \cdot z1\right) + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(z1 \cdot z1\right)} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(z1 \cdot z1\right)} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      3. associate-*r* N/A

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot z1\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      4. remove-double-neg N/A

         \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z1\right)\right)\right)\right)}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(z1\right)\right)\right)\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(z1\right)\right)\right)\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      7. distribute-rgt-neg-out N/A

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z1\right)\right)\right)\right)\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      8. remove-double-neg N/A

         \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{z1}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      9. lower-*.f64 63.3%

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot z1\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto {\left(\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      12. pow-neg N/A

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      13. lower-unsound-pow.f32 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      14. lower-pow.f32 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      15. pow1/2 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\sqrt{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto {\left(\left(\frac{1}{\sqrt{\color{blue}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

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

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      18. lift-fabs.f64 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      19. lower-unsound-/.f64 72.5%

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

   7. Applied rewrites 72.5%

      \[\leadsto {\left(\color{blue}{\left(\frac{1}{\left|z0\right|} \cdot z1\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto {\left(\left(\color{blue}{\frac{1}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      2. frac-2neg N/A

         \[\leadsto {\left(\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left|z0\right|\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\left|z0\right|\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      4. frac-2neg N/A

         \[\leadsto {\left(\left(\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|z0\right|\right)\right)\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto {\left(\left(\frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|z0\right|\right)\right)\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      6. metadata-eval N/A

         \[\leadsto {\left(\left(\frac{\color{blue}{\sqrt{1}}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|z0\right|\right)\right)\right)} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      7. remove-double-neg N/A

         \[\leadsto {\left(\left(\frac{\sqrt{1}}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      8. lift-fabs.f64 N/A

         \[\leadsto {\left(\left(\frac{\sqrt{1}}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

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

         \[\leadsto {\left(\left(\frac{\sqrt{1}}{\color{blue}{\sqrt{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      10. sqrt-undiv N/A

          \[\leadsto {\left(\left(\color{blue}{\sqrt{\frac{1}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto {\left(\left(\color{blue}{\sqrt{\frac{1}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto {\left(\left(\sqrt{\color{blue}{\frac{1}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      13. lower-*.f64 62.6%

          \[\leadsto {\left(\left(\sqrt{\frac{1}{\color{blue}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto {\left(\left(\color{blue}{\sqrt{\frac{1}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|z0\right|\right|\\ \mathbf{if}\;\left|z0\right| \leq \frac{6338253001141147}{79228162514264337593543950336}:\\ \;\;\;\;{\left(\left|z0\right| \cdot \left|z0\right| + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{z1}{t\_0} \cdot z1 + t\_0\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {t\_0}^{\left(z2 \cdot \frac{1}{2}\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (let* ((t_0 (fabs (fabs z0))))
  (if (<= (fabs z0) 6338253001141147/79228162514264337593543950336)
    (pow (+ (* (fabs z0) (fabs z0)) (* z1 z1)) (* 1/2 z2))
    (*
     (pow (+ (* (/ z1 t_0) z1) t_0) (* z2 1/2))
     (pow t_0 (* z2 1/2))))))

C

double code(double z0, double z1, double z2) {
	double t_0 = fabs(fabs(z0));
	double tmp;
	if (fabs(z0) <= 8e-14) {
		tmp = pow(((fabs(z0) * fabs(z0)) + (z1 * z1)), (0.5 * z2));
	} else {
		tmp = pow((((z1 / t_0) * z1) + t_0), (z2 * 0.5)) * pow(t_0, (z2 * 0.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(abs(z0))
    if (abs(z0) <= 8d-14) then
        tmp = ((abs(z0) * abs(z0)) + (z1 * z1)) ** (0.5d0 * z2)
    else
        tmp = ((((z1 / t_0) * z1) + t_0) ** (z2 * 0.5d0)) * (t_0 ** (z2 * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1, double z2) {
	double t_0 = Math.abs(Math.abs(z0));
	double tmp;
	if (Math.abs(z0) <= 8e-14) {
		tmp = Math.pow(((Math.abs(z0) * Math.abs(z0)) + (z1 * z1)), (0.5 * z2));
	} else {
		tmp = Math.pow((((z1 / t_0) * z1) + t_0), (z2 * 0.5)) * Math.pow(t_0, (z2 * 0.5));
	}
	return tmp;
}

Python

def code(z0, z1, z2):
	t_0 = math.fabs(math.fabs(z0))
	tmp = 0
	if math.fabs(z0) <= 8e-14:
		tmp = math.pow(((math.fabs(z0) * math.fabs(z0)) + (z1 * z1)), (0.5 * z2))
	else:
		tmp = math.pow((((z1 / t_0) * z1) + t_0), (z2 * 0.5)) * math.pow(t_0, (z2 * 0.5))
	return tmp

Julia

function code(z0, z1, z2)
	t_0 = abs(abs(z0))
	tmp = 0.0
	if (abs(z0) <= 8e-14)
		tmp = Float64(Float64(abs(z0) * abs(z0)) + Float64(z1 * z1)) ^ Float64(0.5 * z2);
	else
		tmp = Float64((Float64(Float64(Float64(z1 / t_0) * z1) + t_0) ^ Float64(z2 * 0.5)) * (t_0 ^ Float64(z2 * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2)
	t_0 = abs(abs(z0));
	tmp = 0.0;
	if (abs(z0) <= 8e-14)
		tmp = ((abs(z0) * abs(z0)) + (z1 * z1)) ^ (0.5 * z2);
	else
		tmp = ((((z1 / t_0) * z1) + t_0) ^ (z2 * 0.5)) * (t_0 ^ (z2 * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_] := Block[{t$95$0 = N[Abs[N[Abs[z0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 6338253001141147/79228162514264337593543950336], N[Power[N[(N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] + N[(z1 * z1), $MachinePrecision]), $MachinePrecision], N[(1/2 * z2), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(N[(N[(z1 / t$95$0), $MachinePrecision] * z1), $MachinePrecision] + t$95$0), $MachinePrecision], N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 8e-14

   1. Initial program 76.3%

      \[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

   if 8e-14 < z0

   1. Initial program 76.3%

      \[{\left(z0 \cdot z0 + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]
   2. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto {\left(\color{blue}{{\left(z0 \cdot z0\right)}^{1}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      2. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{1 - -1}}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      4. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{1 - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      5. metadata-eval N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{\frac{1}{2} \cdot 2} - \left(\mathsf{neg}\left(1\right)\right)}{2}\right)} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      6. sub-to-fraction N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      7. pow-sub N/A

         \[\leadsto {\left(\color{blue}{\frac{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      8. lower-unsound-pow.f32 N/A

         \[\leadsto {\left(\frac{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      9. lower-pow.f32 N/A

         \[\leadsto {\left(\frac{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      10. pow1/2 N/A

          \[\leadsto {\left(\frac{\color{blue}{\sqrt{z0 \cdot z0}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\sqrt{\color{blue}{z0 \cdot z0}}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

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

          \[\leadsto {\left(\frac{\color{blue}{\left|z0\right|}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      13. lower-unsound-/.f64 N/A

          \[\leadsto {\left(\color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      14. lower-fabs.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{\left|z0\right|}}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto {\left(\frac{\left|z0\right|}{\color{blue}{{\left(z0 \cdot z0\right)}^{\left(\frac{\mathsf{neg}\left(1\right)}{2}\right)}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      16. metadata-eval N/A

          \[\leadsto {\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      17. metadata-eval 76.3%

          \[\leadsto {\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\color{blue}{\frac{-1}{2}}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

   3. Applied rewrites 76.3%

      \[\leadsto {\left(\color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} + z1 \cdot z1\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} + z1 \cdot z1\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      3. +-commutative N/A

         \[\leadsto {\color{blue}{\left(z1 \cdot z1 + \frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto {\left(z1 \cdot z1 + \color{blue}{\frac{\left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      5. add-to-fraction N/A

         \[\leadsto {\color{blue}{\left(\frac{\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      6. mult-flip N/A

         \[\leadsto {\color{blue}{\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \frac{1}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}\right)}}^{\left(\frac{1}{2} \cdot z2\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      8. pow-flip N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{{\left(z0 \cdot z0\right)}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      9. metadata-eval N/A

         \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot {\left(z0 \cdot z0\right)}^{\color{blue}{\frac{1}{2}}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      10. pow1/2 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\sqrt{z0 \cdot z0}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \sqrt{\color{blue}{z0 \cdot z0}}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

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

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\left|z0\right|}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      13. lift-fabs.f64 N/A

          \[\leadsto {\left(\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right) \cdot \color{blue}{\left|z0\right|}\right)}^{\left(\frac{1}{2} \cdot z2\right)} \]

      14. unpow-prod-down N/A

          \[\leadsto \color{blue}{{\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)} \cdot {\left(\left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

      15. lower-unsound-*.f64 N/A

          \[\leadsto \color{blue}{{\left(\left(z1 \cdot z1\right) \cdot {\left(z0 \cdot z0\right)}^{\frac{-1}{2}} + \left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)} \cdot {\left(\left|z0\right|\right)}^{\left(\frac{1}{2} \cdot z2\right)}} \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(z1 \cdot z1\right) + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(z1 \cdot z1\right)} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(z1 \cdot z1\right)} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      3. associate-*r* N/A

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot z1\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      4. remove-double-neg N/A

         \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z1\right)\right)\right)\right)}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(z1\right)\right)\right)\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(\mathsf{neg}\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(z1\right)\right)\right)\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      7. distribute-rgt-neg-out N/A

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z1\right)\right)\right)\right)\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      8. remove-double-neg N/A

         \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot \color{blue}{z1}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      9. lower-*.f64 63.3%

         \[\leadsto {\left(\color{blue}{\left({\left(z0 \cdot z0\right)}^{\frac{-1}{2}} \cdot z1\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto {\left(\left(\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{-1}{2}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto {\left(\left({\left(z0 \cdot z0\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      12. pow-neg N/A

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      13. lower-unsound-pow.f32 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      14. lower-pow.f32 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{{\left(z0 \cdot z0\right)}^{\frac{1}{2}}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      15. pow1/2 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\sqrt{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto {\left(\left(\frac{1}{\sqrt{\color{blue}{z0 \cdot z0}}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

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

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      18. lift-fabs.f64 N/A

          \[\leadsto {\left(\left(\frac{1}{\color{blue}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      19. lower-unsound-/.f64 72.5%

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{\left|z0\right|}} \cdot z1\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

   7. Applied rewrites 72.5%

      \[\leadsto {\left(\color{blue}{\left(\frac{1}{\left|z0\right|} \cdot z1\right) \cdot z1} + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {\left(\color{blue}{\left(\frac{1}{\left|z0\right|} \cdot z1\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\color{blue}{\left(z1 \cdot \frac{1}{\left|z0\right|}\right)} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(\left(z1 \cdot \color{blue}{\frac{1}{\left|z0\right|}}\right) \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      4. mult-flip-rev N/A

         \[\leadsto {\left(\color{blue}{\frac{z1}{\left|z0\right|}} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

      5. lower-/.f64 72.5%

         \[\leadsto {\left(\color{blue}{\frac{z1}{\left|z0\right|}} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]

   9. Applied rewrites 72.5%

      \[\leadsto {\left(\color{blue}{\frac{z1}{\left|z0\right|}} \cdot z1 + \left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \cdot {\left(\left|z0\right|\right)}^{\left(z2 \cdot \frac{1}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0 z1 z2)
  :name "(pow (+ (* z0 z0) (* z1 z1)) (* 1/2 z2))"
  :precision binary64
  (pow (+ (* z0 z0) (* z1 z1)) (* 1/2 z2)))