(sqrt (* (* (+ z0 z0) z1) (- (- z2 (* (/ z3 z4) (+ z3 z3))) (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5)))))

Percentage Accurate: 53.4% → 64.9%

Time: 30.8s

Alternatives: 7

Speedup: 1.2×

Specification

Math

\[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (sqrt
 (*
  (* (+ z0 z0) z1)
  (-
   (- z2 (* (/ z3 z4) (+ z3 z3)))
   (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5))))))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
}

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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    code = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))))
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return Math.sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
}

Python

def code(z0, z1, z2, z3, z4, z5):
	return math.sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))))

Julia

function code(z0, z1, z2, z3, z4, z5)
	return sqrt(Float64(Float64(Float64(z0 + z0) * z1) * Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(z3 + z3))) - Float64(Float64(z0 * Float64(Float64(z3 / z4) * Float64(z3 / z4))) * Float64(z1 - z5)))))
end

MATLAB

function tmp = code(z0, z1, z2, z3, z4, z5)
	tmp = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := N[Sqrt[N[(N[(N[(z0 + z0), $MachinePrecision] * z1), $MachinePrecision] * N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(z3 + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z0 * N[(N[(z3 / z4), $MachinePrecision] * N[(z3 / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z1 - z5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.4% accurate, 1.0× speedup

Math

\[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (sqrt
 (*
  (* (+ z0 z0) z1)
  (-
   (- z2 (* (/ z3 z4) (+ z3 z3)))
   (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5))))))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
}

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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    code = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))))
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return Math.sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
}

Python

def code(z0, z1, z2, z3, z4, z5):
	return math.sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))))

Julia

function code(z0, z1, z2, z3, z4, z5)
	return sqrt(Float64(Float64(Float64(z0 + z0) * z1) * Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(z3 + z3))) - Float64(Float64(z0 * Float64(Float64(z3 / z4) * Float64(z3 / z4))) * Float64(z1 - z5)))))
end

MATLAB

function tmp = code(z0, z1, z2, z3, z4, z5)
	tmp = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := N[Sqrt[N[(N[(N[(z0 + z0), $MachinePrecision] * z1), $MachinePrecision] * N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(z3 + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z0 * N[(N[(z3 / z4), $MachinePrecision] * N[(z3 / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z1 - z5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 64.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{z3}{z4} \cdot z0\\ \mathbf{if}\;\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \leq 1000000000000000017652801462756379714374878780719864776839443139119744823869255243069012222883470359078822072829219411228534934402712624705615450492327979456500795456339201761949451160807447294527656222743617592048849967890105831362861792425329827928397252374398383022243308510390698430058459037696:\\ \;\;\;\;\sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \left(z5 - z1\right) \cdot t\_0\right)\right), z1, z0, \left(-z1\right), z0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(t\_0 \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (let* ((t_0 (* (/ z3 z4) z0)))
  (if (<=
       (*
        (* (+ z0 z0) z1)
        (-
         (- z2 (* (/ z3 z4) (+ z3 z3)))
         (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5))))
       1000000000000000017652801462756379714374878780719864776839443139119744823869255243069012222883470359078822072829219411228534934402712624705615450492327979456500795456339201761949451160807447294527656222743617592048849967890105831362861792425329827928397252374398383022243308510390698430058459037696)
    (sqrt
     (304-z0z1z2z3z4
      (- z2 (* (/ z3 z4) (- (+ z3 z3) (* (- z5 z1) t_0))))
      z1
      z0
      (- z1)
      z0))
    (sqrt
     (*
      (*
       (- z2 (* (/ z3 z4) (+ (+ (* t_0 (- z1 z5)) z3) z3)))
       (+ z0 z0))
      z1)))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5)))) < 1e297

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \left(\left(z0 + z0\right) \cdot z1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(z1 \cdot \left(z0 + z0\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \left(z1 \cdot \color{blue}{\left(z0 + z0\right)}\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(z1 \cdot z0 + z1 \cdot z0\right)}} \]

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

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(z1 \cdot z0 - \left(\mathsf{neg}\left(z1\right)\right) \cdot z0\right)}} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right), z1, z0, \left(\mathsf{neg}\left(z1\right)\right), z0\right)}} \]

   3. Applied rewrites 48.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right), z1, z0, \left(-z1\right), z0\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right), z1, z0, \left(-z1\right), z0\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}\right)}\right), z1, z0, \left(-z1\right), z0\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}}\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right)} \cdot \frac{z3}{z4}\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) + \color{blue}{\left(z1 - z5\right) \cdot \left(z0 \cdot \frac{z3}{z4}\right)}\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

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

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(z3 + z3\right) - \left(\mathsf{neg}\left(\left(z1 - z5\right)\right)\right) \cdot \left(z0 \cdot \frac{z3}{z4}\right)\right)}\right), z1, z0, \left(-z1\right), z0\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(z3 + z3\right) - \left(\mathsf{neg}\left(\left(z1 - z5\right)\right)\right) \cdot \left(z0 \cdot \frac{z3}{z4}\right)\right)}\right), z1, z0, \left(-z1\right), z0\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \left(\mathsf{neg}\left(\color{blue}{\left(z1 - z5\right)}\right)\right) \cdot \left(z0 \cdot \frac{z3}{z4}\right)\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      9. sub-negate-rev N/A

         \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \color{blue}{\left(z5 - z1\right)} \cdot \left(z0 \cdot \frac{z3}{z4}\right)\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \left(z5 - z1\right) \cdot \color{blue}{\left(\frac{z3}{z4} \cdot z0\right)}\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \color{blue}{\left(z5 - z1\right) \cdot \left(\frac{z3}{z4} \cdot z0\right)}\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \color{blue}{\left(z5 - z1\right)} \cdot \left(\frac{z3}{z4} \cdot z0\right)\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

      13. lower-*.f64 52.8%

          \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(z3 + z3\right) - \left(z5 - z1\right) \cdot \color{blue}{\left(\frac{z3}{z4} \cdot z0\right)}\right)\right), z1, z0, \left(-z1\right), z0\right)} \]

   5. Applied rewrites 52.8%

      \[\leadsto \sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(z3 + z3\right) - \left(z5 - z1\right) \cdot \left(\frac{z3}{z4} \cdot z0\right)\right)}\right), z1, z0, \left(-z1\right), z0\right)} \]

   if 1e297 < (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5))))

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   3. Applied rewrites 55.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \color{blue}{\left(z3 + z3\right)}\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      5. lower-+.f64 55.8%

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right)} + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\frac{z3}{z4} \cdot \left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(z0 \cdot \left(z1 - z5\right)\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      12. lower-*.f64 59.5%

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right)} \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. Applied rewrites 59.5%

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}\\ t_1 := \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}\\ \mathbf{if}\;t\_0 \leq 1000000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 199999999999999982405111001914463627825705729939051460364922737117355163153802565541919878198424069508213948681199740222346696327168:\\ \;\;\;\;\sqrt{\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z1 \cdot \left(z0 + z0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (let* ((t_0
        (sqrt
         (*
          (* (+ z0 z0) z1)
          (-
           (- z2 (* (/ z3 z4) (+ z3 z3)))
           (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5))))))
       (t_1
        (sqrt
         (*
          (*
           (-
            z2
            (*
             (/ z3 z4)
             (+ (+ (* (* (/ z3 z4) z0) (- z1 z5)) z3) z3)))
           (+ z0 z0))
          z1))))
  (if (<= t_0 1000000000000000)
    t_1
    (if (<=
         t_0
         199999999999999982405111001914463627825705729939051460364922737117355163153802565541919878198424069508213948681199740222346696327168)
      (sqrt
       (*
        (-
         z2
         (* (/ z3 z4) (+ (* (* (- z1 z5) z0) (/ z3 z4)) (+ z3 z3))))
        (* z1 (+ z0 z0))))
      t_1))))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	double t_0 = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
	double t_1 = sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1));
	double tmp;
	if (t_0 <= 1e+15) {
		tmp = t_1;
	} else if (t_0 <= 2e+131) {
		tmp = sqrt(((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z1 * (z0 + z0))));
	} else {
		tmp = t_1;
	}
	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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))))
    t_1 = sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1))
    if (t_0 <= 1d+15) then
        tmp = t_1
    else if (t_0 <= 2d+131) then
        tmp = sqrt(((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z1 * (z0 + z0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	double t_0 = Math.sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
	double t_1 = Math.sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1));
	double tmp;
	if (t_0 <= 1e+15) {
		tmp = t_1;
	} else if (t_0 <= 2e+131) {
		tmp = Math.sqrt(((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z1 * (z0 + z0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z0, z1, z2, z3, z4, z5):
	t_0 = math.sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))))
	t_1 = math.sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1))
	tmp = 0
	if t_0 <= 1e+15:
		tmp = t_1
	elif t_0 <= 2e+131:
		tmp = math.sqrt(((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z1 * (z0 + z0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(z0, z1, z2, z3, z4, z5)
	t_0 = sqrt(Float64(Float64(Float64(z0 + z0) * z1) * Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(z3 + z3))) - Float64(Float64(z0 * Float64(Float64(z3 / z4) * Float64(z3 / z4))) * Float64(z1 - z5)))))
	t_1 = sqrt(Float64(Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(Float64(Float64(Float64(Float64(z3 / z4) * z0) * Float64(z1 - z5)) + z3) + z3))) * Float64(z0 + z0)) * z1))
	tmp = 0.0
	if (t_0 <= 1e+15)
		tmp = t_1;
	elseif (t_0 <= 2e+131)
		tmp = sqrt(Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(Float64(Float64(Float64(z1 - z5) * z0) * Float64(z3 / z4)) + Float64(z3 + z3)))) * Float64(z1 * Float64(z0 + z0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2, z3, z4, z5)
	t_0 = sqrt((((z0 + z0) * z1) * ((z2 - ((z3 / z4) * (z3 + z3))) - ((z0 * ((z3 / z4) * (z3 / z4))) * (z1 - z5)))));
	t_1 = sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1));
	tmp = 0.0;
	if (t_0 <= 1e+15)
		tmp = t_1;
	elseif (t_0 <= 2e+131)
		tmp = sqrt(((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z1 * (z0 + z0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := Block[{t$95$0 = N[Sqrt[N[(N[(N[(z0 + z0), $MachinePrecision] * z1), $MachinePrecision] * N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(z3 + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z0 * N[(N[(z3 / z4), $MachinePrecision] * N[(z3 / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z1 - z5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(N[(N[(N[(N[(z3 / z4), $MachinePrecision] * z0), $MachinePrecision] * N[(z1 - z5), $MachinePrecision]), $MachinePrecision] + z3), $MachinePrecision] + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1000000000000000], t$95$1, If[LessEqual[t$95$0, 199999999999999982405111001914463627825705729939051460364922737117355163153802565541919878198424069508213948681199740222346696327168], N[Sqrt[N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(N[(N[(N[(z1 - z5), $MachinePrecision] * z0), $MachinePrecision] * N[(z3 / z4), $MachinePrecision]), $MachinePrecision] + N[(z3 + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z1 * N[(z0 + z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5))))) < 1e15 or 1.9999999999999998e131 < (sqrt.f64 (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5)))))

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   3. Applied rewrites 55.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \color{blue}{\left(z3 + z3\right)}\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      5. lower-+.f64 55.8%

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right)} + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\frac{z3}{z4} \cdot \left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(z0 \cdot \left(z1 - z5\right)\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      12. lower-*.f64 59.5%

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right)} \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. Applied rewrites 59.5%

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   if 1e15 < (sqrt.f64 (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5))))) < 1.9999999999999998e131

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \left(\left(z0 + z0\right) \cdot z1\right)}} \]

      3. lower-*.f64 53.4%

         \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \left(\left(z0 + z0\right) \cdot z1\right)}} \]

   3. Applied rewrites 55.5%

      \[\leadsto \sqrt{\color{blue}{\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z1 \cdot \left(z0 + z0\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \leq 5000000000000000111755861797384299667549204650486879780239418214450132430121171797988101755921550297505076285418812476851459272474624:\\ \;\;\;\;\sqrt{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right), z1, z0, \left(-z1\right), z0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (if (<=
     (sqrt
      (*
       (* (+ z0 z0) z1)
       (-
        (- z2 (* (/ z3 z4) (+ z3 z3)))
        (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5)))))
     5000000000000000111755861797384299667549204650486879780239418214450132430121171797988101755921550297505076285418812476851459272474624)
  (sqrt
   (304-z0z1z2z3z4
    (- z2 (* (/ z3 z4) (+ (* (* (- z1 z5) z0) (/ z3 z4)) (+ z3 z3))))
    z1
    z0
    (- z1)
    z0))
  (sqrt
   (*
    (*
     (- z2 (* (/ z3 z4) (+ (+ (* (* (/ z3 z4) z0) (- z1 z5)) z3) z3)))
     (+ z0 z0))
    z1))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5))))) < 5.0000000000000001e132

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \left(\left(z0 + z0\right) \cdot z1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(z1 \cdot \left(z0 + z0\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \left(z1 \cdot \color{blue}{\left(z0 + z0\right)}\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(z1 \cdot z0 + z1 \cdot z0\right)}} \]

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

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot \color{blue}{\left(z1 \cdot z0 - \left(\mathsf{neg}\left(z1\right)\right) \cdot z0\right)}} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right), z1, z0, \left(\mathsf{neg}\left(z1\right)\right), z0\right)}} \]

   3. Applied rewrites 48.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right), z1, z0, \left(-z1\right), z0\right)}} \]

   if 5.0000000000000001e132 < (sqrt.f64 (*.f64 (*.f64 (+.f64 z0 z0) z1) (-.f64 (-.f64 z2 (*.f64 (/.f64 z3 z4) (+.f64 z3 z3))) (*.f64 (*.f64 z0 (*.f64 (/.f64 z3 z4) (/.f64 z3 z4))) (-.f64 z1 z5)))))

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   3. Applied rewrites 55.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \color{blue}{\left(z3 + z3\right)}\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      5. lower-+.f64 55.8%

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right)} + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\frac{z3}{z4} \cdot \left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(z0 \cdot \left(z1 - z5\right)\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      12. lower-*.f64 59.5%

          \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right)} \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. Applied rewrites 59.5%

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 59.5% accurate, 1.2× speedup

Math

\[\sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (sqrt
 (*
  (*
   (- z2 (* (/ z3 z4) (+ (+ (* (* (/ z3 z4) z0) (- z1 z5)) z3) z3)))
   (+ z0 z0))
  z1)))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1));
}

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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    code = sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1))
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return Math.sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1));
}

Python

def code(z0, z1, z2, z3, z4, z5):
	return math.sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1))

Julia

function code(z0, z1, z2, z3, z4, z5)
	return sqrt(Float64(Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(Float64(Float64(Float64(Float64(z3 / z4) * z0) * Float64(z1 - z5)) + z3) + z3))) * Float64(z0 + z0)) * z1))
end

MATLAB

function tmp = code(z0, z1, z2, z3, z4, z5)
	tmp = sqrt((((z2 - ((z3 / z4) * (((((z3 / z4) * z0) * (z1 - z5)) + z3) + z3))) * (z0 + z0)) * z1));
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := N[Sqrt[N[(N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(N[(N[(N[(N[(z3 / z4), $MachinePrecision] * z0), $MachinePrecision] * N[(z1 - z5), $MachinePrecision]), $MachinePrecision] + z3), $MachinePrecision] + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

3. Applied rewrites 55.8%

   \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   2. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \color{blue}{\left(z3 + z3\right)}\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   3. associate-+r+ N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   4. lower-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. lower-+.f64 55.8%

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + z3\right)} + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\frac{z3}{z4} \cdot \left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   9. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\frac{z3}{z4} \cdot \color{blue}{\left(z0 \cdot \left(z1 - z5\right)\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   10. associate-*r* N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   11. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right)} + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   12. lower-*.f64 59.5%

       \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\color{blue}{\left(\frac{z3}{z4} \cdot z0\right)} \cdot \left(z1 - z5\right) + z3\right) + z3\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

5. Applied rewrites 59.5%

   \[\leadsto \sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \color{blue}{\left(\left(\left(\frac{z3}{z4} \cdot z0\right) \cdot \left(z1 - z5\right) + z3\right) + z3\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
6. Add Preprocessing

Alternative 5: 55.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -64999999999999996793668510548615010169459801421748008099328347828280602319944494758993776569724065175462201349583033252647656195072525976144106252024887810771477419024973824:\\ \;\;\;\;\sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(z5 - z1\right) \cdot \left(z3 \cdot z0\right)}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (if (<=
     z0
     -64999999999999996793668510548615010169459801421748008099328347828280602319944494758993776569724065175462201349583033252647656195072525976144106252024887810771477419024973824)
  (sqrt
   (*
    (*
     (-
      z2
      (*
       z3
       (/ (- (* (+ z3 z3) z4) (* (- z5 z1) (* z3 z0))) (* z4 z4))))
     (+ z0 z0))
    z1))
  (sqrt
   (*
    (*
     (- z2 (* (/ z3 z4) (+ (* (* (- z1 z5) z0) (/ z3 z4)) (+ z3 z3))))
     (+ z0 z0))
    z1))))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	double tmp;
	if (z0 <= -6.5e+172) {
		tmp = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1));
	} else {
		tmp = sqrt((((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z0 + z0)) * z1));
	}
	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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    real(8) :: tmp
    if (z0 <= (-6.5d+172)) then
        tmp = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1))
    else
        tmp = sqrt((((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z0 + z0)) * z1))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	double tmp;
	if (z0 <= -6.5e+172) {
		tmp = Math.sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1));
	} else {
		tmp = Math.sqrt((((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z0 + z0)) * z1));
	}
	return tmp;
}

Python

def code(z0, z1, z2, z3, z4, z5):
	tmp = 0
	if z0 <= -6.5e+172:
		tmp = math.sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1))
	else:
		tmp = math.sqrt((((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z0 + z0)) * z1))
	return tmp

Julia

function code(z0, z1, z2, z3, z4, z5)
	tmp = 0.0
	if (z0 <= -6.5e+172)
		tmp = sqrt(Float64(Float64(Float64(z2 - Float64(z3 * Float64(Float64(Float64(Float64(z3 + z3) * z4) - Float64(Float64(z5 - z1) * Float64(z3 * z0))) / Float64(z4 * z4)))) * Float64(z0 + z0)) * z1));
	else
		tmp = sqrt(Float64(Float64(Float64(z2 - Float64(Float64(z3 / z4) * Float64(Float64(Float64(Float64(z1 - z5) * z0) * Float64(z3 / z4)) + Float64(z3 + z3)))) * Float64(z0 + z0)) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2, z3, z4, z5)
	tmp = 0.0;
	if (z0 <= -6.5e+172)
		tmp = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1));
	else
		tmp = sqrt((((z2 - ((z3 / z4) * ((((z1 - z5) * z0) * (z3 / z4)) + (z3 + z3)))) * (z0 + z0)) * z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := If[LessEqual[z0, -64999999999999996793668510548615010169459801421748008099328347828280602319944494758993776569724065175462201349583033252647656195072525976144106252024887810771477419024973824], N[Sqrt[N[(N[(N[(z2 - N[(z3 * N[(N[(N[(N[(z3 + z3), $MachinePrecision] * z4), $MachinePrecision] - N[(N[(z5 - z1), $MachinePrecision] * N[(z3 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(z2 - N[(N[(z3 / z4), $MachinePrecision] * N[(N[(N[(N[(z1 - z5), $MachinePrecision] * z0), $MachinePrecision] * N[(z3 / z4), $MachinePrecision]), $MachinePrecision] + N[(z3 + z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -6.4999999999999997e172

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   3. Applied rewrites 55.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right) \cdot \frac{z3}{z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}\right)} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \color{blue}{\frac{z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \color{blue}{\frac{\left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      8. add-to-fraction N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4}} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4} \cdot \color{blue}{\frac{z3}{z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      10. frac-times N/A

          \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{\color{blue}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      14. lower-+.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right)} \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\color{blue}{\left(z3 + z3\right) \cdot z4} + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      16. lower-*.f64 41.5%

          \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\left(z3 + z3\right) \cdot z4 + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. Applied rewrites 41.5%

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{z3 \cdot \left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      6. lower-/.f64 43.7%

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \color{blue}{\frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   7. Applied rewrites 43.7%

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(\left(z5 - z1\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(\left(z5 - z1\right) \cdot z0\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(\left(z5 - z1\right) \cdot z0\right)} \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(z5 - z1\right) \cdot \left(z0 \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(z5 - z1\right) \cdot \left(z0 \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(z5 - z1\right) \cdot \color{blue}{\left(z3 \cdot z0\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

      6. lower-*.f64 46.0%

         \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(z5 - z1\right) \cdot \color{blue}{\left(z3 \cdot z0\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   9. Applied rewrites 46.0%

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(z5 - z1\right) \cdot \left(z3 \cdot z0\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   if -6.4999999999999997e172 < z0

   1. Initial program 53.4%

      \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   3. Applied rewrites 55.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 46.0% accurate, 1.2× speedup

Math

\[\sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(z5 - z1\right) \cdot \left(z3 \cdot z0\right)}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (sqrt
 (*
  (*
   (-
    z2
    (* z3 (/ (- (* (+ z3 z3) z4) (* (- z5 z1) (* z3 z0))) (* z4 z4))))
   (+ z0 z0))
  z1)))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1));
}

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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    code = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1))
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return Math.sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1));
}

Python

def code(z0, z1, z2, z3, z4, z5):
	return math.sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1))

Julia

function code(z0, z1, z2, z3, z4, z5)
	return sqrt(Float64(Float64(Float64(z2 - Float64(z3 * Float64(Float64(Float64(Float64(z3 + z3) * z4) - Float64(Float64(z5 - z1) * Float64(z3 * z0))) / Float64(z4 * z4)))) * Float64(z0 + z0)) * z1))
end

MATLAB

function tmp = code(z0, z1, z2, z3, z4, z5)
	tmp = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - ((z5 - z1) * (z3 * z0))) / (z4 * z4)))) * (z0 + z0)) * z1));
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := N[Sqrt[N[(N[(N[(z2 - N[(z3 * N[(N[(N[(N[(z3 + z3), $MachinePrecision] * z4), $MachinePrecision] - N[(N[(z5 - z1), $MachinePrecision] * N[(z3 * z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

3. Applied rewrites 55.8%

   \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right) \cdot \frac{z3}{z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   3. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   4. +-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}\right)} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   6. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \color{blue}{\frac{z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   7. associate-*r/ N/A

      \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \color{blue}{\frac{\left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   8. add-to-fraction N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4}} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4} \cdot \color{blue}{\frac{z3}{z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   10. frac-times N/A

       \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{\color{blue}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   12. lower-/.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   13. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   14. lower-+.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right)} \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   15. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\color{blue}{\left(z3 + z3\right) \cdot z4} + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   16. lower-*.f64 41.5%

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\left(z3 + z3\right) \cdot z4 + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

5. Applied rewrites 41.5%

   \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{z3 \cdot \left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   4. associate-/l* N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   6. lower-/.f64 43.7%

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \color{blue}{\frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

7. Applied rewrites 43.7%

   \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(\left(z5 - z1\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(\left(z5 - z1\right) \cdot z0\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(\left(z5 - z1\right) \cdot z0\right)} \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(z5 - z1\right) \cdot \left(z0 \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(z5 - z1\right) \cdot \left(z0 \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(z5 - z1\right) \cdot \color{blue}{\left(z3 \cdot z0\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   6. lower-*.f64 46.0%

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(z5 - z1\right) \cdot \color{blue}{\left(z3 \cdot z0\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

9. Applied rewrites 46.0%

   \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \color{blue}{\left(z5 - z1\right) \cdot \left(z3 \cdot z0\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
10. Add Preprocessing

Alternative 7: 43.7% accurate, 1.2× speedup

Math

\[\sqrt{\left(\left(z2 - z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(\left(z5 - z1\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

FPCore

(FPCore (z0 z1 z2 z3 z4 z5)
  :precision binary64
  (sqrt
 (*
  (*
   (-
    z2
    (* z3 (/ (- (* (+ z3 z3) z4) (* (* (- z5 z1) z0) z3)) (* z4 z4))))
   (+ z0 z0))
  z1)))

C

double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - (((z5 - z1) * z0) * z3)) / (z4 * z4)))) * (z0 + z0)) * z1));
}

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, z3, z4, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z5
    code = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - (((z5 - z1) * z0) * z3)) / (z4 * z4)))) * (z0 + z0)) * z1))
end function

Java

public static double code(double z0, double z1, double z2, double z3, double z4, double z5) {
	return Math.sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - (((z5 - z1) * z0) * z3)) / (z4 * z4)))) * (z0 + z0)) * z1));
}

Python

def code(z0, z1, z2, z3, z4, z5):
	return math.sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - (((z5 - z1) * z0) * z3)) / (z4 * z4)))) * (z0 + z0)) * z1))

Julia

function code(z0, z1, z2, z3, z4, z5)
	return sqrt(Float64(Float64(Float64(z2 - Float64(z3 * Float64(Float64(Float64(Float64(z3 + z3) * z4) - Float64(Float64(Float64(z5 - z1) * z0) * z3)) / Float64(z4 * z4)))) * Float64(z0 + z0)) * z1))
end

MATLAB

function tmp = code(z0, z1, z2, z3, z4, z5)
	tmp = sqrt((((z2 - (z3 * ((((z3 + z3) * z4) - (((z5 - z1) * z0) * z3)) / (z4 * z4)))) * (z0 + z0)) * z1));
end

Wolfram

code[z0_, z1_, z2_, z3_, z4_, z5_] := N[Sqrt[N[(N[(N[(z2 - N[(z3 * N[(N[(N[(N[(z3 + z3), $MachinePrecision] * z4), $MachinePrecision] - N[(N[(N[(z5 - z1), $MachinePrecision] * z0), $MachinePrecision] * z3), $MachinePrecision]), $MachinePrecision] / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sqrt{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot z1\right)} \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\color{blue}{\left(z0 + z0\right) \cdot \left(z1 \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(z0 + z0\right) \cdot \color{blue}{\left(\left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right) \cdot z1\right)}} \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 + z0\right) \cdot \left(\left(z2 - \frac{z3}{z4} \cdot \left(z3 + z3\right)\right) - \left(z0 \cdot \left(\frac{z3}{z4} \cdot \frac{z3}{z4}\right)\right) \cdot \left(z1 - z5\right)\right)\right) \cdot z1}} \]

3. Applied rewrites 55.8%

   \[\leadsto \sqrt{\color{blue}{\left(\left(z2 - \frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)\right) \cdot \left(z0 + z0\right)\right) \cdot z1}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{z3}{z4} \cdot \left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right) \cdot \frac{z3}{z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   3. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4} + \left(z3 + z3\right)\right)} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   4. +-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}\right)} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot \frac{z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   6. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \left(\left(z1 - z5\right) \cdot z0\right) \cdot \color{blue}{\frac{z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   7. associate-*r/ N/A

      \[\leadsto \sqrt{\left(\left(z2 - \left(\left(z3 + z3\right) + \color{blue}{\frac{\left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4}}\right) \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   8. add-to-fraction N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4}} \cdot \frac{z3}{z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4} \cdot \color{blue}{\frac{z3}{z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   10. frac-times N/A

       \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{\color{blue}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   12. lower-/.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   13. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   14. lower-+.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right)} \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   15. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\color{blue}{\left(z3 + z3\right) \cdot z4} + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   16. lower-*.f64 41.5%

       \[\leadsto \sqrt{\left(\left(z2 - \frac{\left(\left(z3 + z3\right) \cdot z4 + \color{blue}{\left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}\right) \cdot z3}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

5. Applied rewrites 41.5%

   \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{\frac{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{\left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right) \cdot z3}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(z2 - \frac{\color{blue}{z3 \cdot \left(\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3\right)}}{z4 \cdot z4}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   4. associate-/l* N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

   6. lower-/.f64 43.7%

      \[\leadsto \sqrt{\left(\left(z2 - z3 \cdot \color{blue}{\frac{\left(z3 + z3\right) \cdot z4 + \left(\left(z1 - z5\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]

7. Applied rewrites 43.7%

   \[\leadsto \sqrt{\left(\left(z2 - \color{blue}{z3 \cdot \frac{\left(z3 + z3\right) \cdot z4 - \left(\left(z5 - z1\right) \cdot z0\right) \cdot z3}{z4 \cdot z4}}\right) \cdot \left(z0 + z0\right)\right) \cdot z1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0 z1 z2 z3 z4 z5)
  :name "(sqrt (* (* (+ z0 z0) z1) (- (- z2 (* (/ z3 z4) (+ z3 z3))) (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5)))))"
  :precision binary64
  (sqrt (* (* (+ z0 z0) z1) (- (- z2 (* (/ z3 z4) (+ z3 z3))) (* (* z0 (* (/ z3 z4) (/ z3 z4))) (- z1 z5))))))