(sqrt (fabs (log (- 1 z0))))

Percentage Accurate: 37.1% → 99.7%

Time: 2.8s

Alternatives: 8

Speedup: 7.7×

Specification

Math

\[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt (fabs (log (- 1.0 z0)))))

C

double code(double z0) {
	return sqrt(fabs(log((1.0 - z0))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs(log((1.0d0 - z0))))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs(Math.log((1.0 - z0))));
}

Python

def code(z0):
	return math.sqrt(math.fabs(math.log((1.0 - z0))))

Julia

function code(z0)
	return sqrt(abs(log(Float64(1.0 - z0))))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs(log((1.0 - z0))));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[Log[N[(1.0 - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 37.1% accurate, 1.0× speedup

Math

\[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt (fabs (log (- 1.0 z0)))))

C

double code(double z0) {
	return sqrt(fabs(log((1.0 - z0))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs(log((1.0d0 - z0))))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs(Math.log((1.0 - z0))));
}

Python

def code(z0):
	return math.sqrt(math.fabs(math.log((1.0 - z0))))

Julia

function code(z0)
	return sqrt(abs(log(Float64(1.0 - z0))))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs(log((1.0 - z0))));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[Log[N[(1.0 - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|\log \left(1 - z0\right)\right|}\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;t\_0 \leq 0.0002:\\ \;\;\;\;\sqrt{\left|\left(\left(-0.3333333333333333 \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - t\_1\right) \cdot 0.5 + \left(t\_0 + t\_1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (sqrt (fabs (log (- 1.0 z0))))) (t_1 (/ 1.0 t_0)))
  (if (<= t_0 0.0002)
    (sqrt
     (fabs (- (* (* (- (* -0.3333333333333333 z0) 0.5) z0) z0) z0)))
    (+ (* (- t_0 t_1) 0.5) (* (+ t_0 t_1) 0.5)))))

C

double code(double z0) {
	double t_0 = sqrt(fabs(log((1.0 - z0))));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = sqrt(fabs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	} else {
		tmp = ((t_0 - t_1) * 0.5) + ((t_0 + t_1) * 0.5);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(abs(log((1.0d0 - z0))))
    t_1 = 1.0d0 / t_0
    if (t_0 <= 0.0002d0) then
        tmp = sqrt(abs(((((((-0.3333333333333333d0) * z0) - 0.5d0) * z0) * z0) - z0)))
    else
        tmp = ((t_0 - t_1) * 0.5d0) + ((t_0 + t_1) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double t_0 = Math.sqrt(Math.abs(Math.log((1.0 - z0))));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = Math.sqrt(Math.abs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	} else {
		tmp = ((t_0 - t_1) * 0.5) + ((t_0 + t_1) * 0.5);
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.sqrt(math.fabs(math.log((1.0 - z0))))
	t_1 = 1.0 / t_0
	tmp = 0
	if t_0 <= 0.0002:
		tmp = math.sqrt(math.fabs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)))
	else:
		tmp = ((t_0 - t_1) * 0.5) + ((t_0 + t_1) * 0.5)
	return tmp

Julia

function code(z0)
	t_0 = sqrt(abs(log(Float64(1.0 - z0))))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (t_0 <= 0.0002)
		tmp = sqrt(abs(Float64(Float64(Float64(Float64(Float64(-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	else
		tmp = Float64(Float64(Float64(t_0 - t_1) * 0.5) + Float64(Float64(t_0 + t_1) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = sqrt(abs(log((1.0 - z0))));
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (t_0 <= 0.0002)
		tmp = sqrt(abs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	else
		tmp = ((t_0 - t_1) * 0.5) + ((t_0 + t_1) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[Log[N[(1.0 - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0002], N[Sqrt[N[Abs[N[(N[(N[(N[(N[(-0.3333333333333333 * z0), $MachinePrecision] - 0.5), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[(N[(t$95$0 - t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[(t$95$0 + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (fabs.f64 (log.f64 (-.f64 #s(literal 1 binary64) z0)))) < 2.0000000000000001e-4

   1. Initial program 37.1%

      \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)}\right|} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)}\right|} \]

      2. lower--.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)\right|} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)\right|} \]

      5. lower-*.f64 68.5%

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(-0.3333333333333333 \cdot z0 - 0.5\right) - 1\right)\right|} \]

   4. Applied rewrites 68.5%

      \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(-0.3333333333333333 \cdot z0 - 0.5\right) - 1\right)}\right|} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)}\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

      3. sub-flip N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right|} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) + -1\right)\right|} \]

      5. distribute-rgt-in N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 + \color{blue}{-1 \cdot z0}\right|} \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 + \left(\mathsf{neg}\left(z0\right)\right)\right|} \]

      7. sub-flip-reverse N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

      8. lower--.f64 N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

      9. lower-*.f64 68.5%

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(-0.3333333333333333 \cdot z0 - 0.5\right)\right) \cdot z0 - z0\right|} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 - z0\right|} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left|\left(\left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

      12. lower-*.f64 68.5%

          \[\leadsto \sqrt{\left|\left(\left(-0.3333333333333333 \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   6. Applied rewrites 68.5%

      \[\leadsto \sqrt{\left|\left(\left(-0.3333333333333333 \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - \color{blue}{z0}\right|} \]

   if 2.0000000000000001e-4 < (sqrt.f64 (fabs.f64 (log.f64 (-.f64 #s(literal 1 binary64) z0))))

   1. Initial program 37.1%

      \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]
   2. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \sqrt{\left|\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - z0\right)\right)\right)\right)}\right|} \]

      2. lift-log.f64 N/A

         \[\leadsto \sqrt{\left|\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\log \left(1 - z0\right)}\right)\right)\right)\right|} \]

      3. neg-log N/A

         \[\leadsto \sqrt{\left|\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{1 - z0}\right)}\right)\right|} \]

      4. neg-log N/A

         \[\leadsto \sqrt{\left|\color{blue}{\log \left(\frac{1}{\frac{1}{1 - z0}}\right)}\right|} \]

      5. *-inverses N/A

         \[\leadsto \sqrt{\left|\log \left(\frac{1}{\frac{\color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}{1 - z0}}\right)\right|} \]

      6. associate-/l/ N/A

         \[\leadsto \sqrt{\left|\log \left(\frac{1}{\color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right) \cdot \left(1 - z0\right)}}}\right)\right|} \]

      7. div-flip-rev N/A

         \[\leadsto \sqrt{\left|\log \color{blue}{\left(\frac{\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right) \cdot \left(1 - z0\right)}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right|} \]

      8. log-div N/A

         \[\leadsto \sqrt{\left|\color{blue}{\log \left(\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right) \cdot \left(1 - z0\right)\right) - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}\right|} \]

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

         \[\leadsto \sqrt{\left|\color{blue}{\log \left(\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right) \cdot \left(1 - z0\right)\right) - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}\right|} \]

      10. lower-unsound-log.f64 N/A

          \[\leadsto \sqrt{\left|\color{blue}{\log \left(\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right) \cdot \left(1 - z0\right)\right)} - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left|\log \color{blue}{\left(\left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right) \cdot \left(1 - z0\right)\right)} - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|} \]

      12. cosh-undef N/A

          \[\leadsto \sqrt{\left|\log \left(\color{blue}{\left(2 \cdot \cosh 0\right)} \cdot \left(1 - z0\right)\right) - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|} \]

      13. cosh-0-rev N/A

          \[\leadsto \sqrt{\left|\log \left(\left(2 \cdot \color{blue}{1}\right) \cdot \left(1 - z0\right)\right) - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left|\log \left(\color{blue}{2} \cdot \left(1 - z0\right)\right) - \log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)\right|} \]

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

          \[\leadsto \sqrt{\left|\log \left(2 \cdot \left(1 - z0\right)\right) - \color{blue}{\log \left(e^{0} + e^{\mathsf{neg}\left(0\right)}\right)}\right|} \]

      16. cosh-undef N/A

          \[\leadsto \sqrt{\left|\log \left(2 \cdot \left(1 - z0\right)\right) - \log \color{blue}{\left(2 \cdot \cosh 0\right)}\right|} \]

   3. Applied rewrites 37.1%

      \[\leadsto \sqrt{\left|\color{blue}{\log \left(2 \cdot \left(1 - z0\right)\right) - \log 2}\right|} \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left|\log \left(2 \cdot \left(1 - z0\right)\right) - \log 2\right|}} \]

      2. pow1/2 N/A

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

      3. remove-double-neg N/A

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

      4. pow-neg N/A

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

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{1}{{\left(\left|\log \left(1 - z0\right)\right|\right)}^{-0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{\left(\left|\log \left(1 - z0\right)\right|\right)}^{\frac{-1}{2}}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{{\left(\left|\log \left(1 - z0\right)\right|\right)}^{\frac{-1}{2}}}} \]

      3. pow-flip N/A

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

      4. metadata-eval N/A

         \[\leadsto {\left(\left|\log \left(1 - z0\right)\right|\right)}^{\color{blue}{\frac{1}{2}}} \]

      5. exp-to-pow N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

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

         \[\leadsto \color{blue}{\cosh \left(\log \left(\left|\log \left(1 - z0\right)\right|\right) \cdot \frac{1}{2}\right) + \sinh \left(\log \left(\left|\log \left(1 - z0\right)\right|\right) \cdot \frac{1}{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\sinh \left(\log \left(\left|\log \left(1 - z0\right)\right|\right) \cdot \frac{1}{2}\right) + \cosh \left(\log \left(\left|\log \left(1 - z0\right)\right|\right) \cdot \frac{1}{2}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(\log \left(\left|\log \left(1 - z0\right)\right|\right) \cdot \frac{1}{2}\right) + \cosh \left(\log \left(\left|\log \left(1 - z0\right)\right|\right) \cdot \frac{1}{2}\right)} \]

   7. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\left(\sqrt{\left|\log \left(1 - z0\right)\right|} - \frac{1}{\sqrt{\left|\log \left(1 - z0\right)\right|}}\right) \cdot 0.5 + \left(\sqrt{\left|\log \left(1 - z0\right)\right|} + \frac{1}{\sqrt{\left|\log \left(1 - z0\right)\right|}}\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|\log \left(1 - z0\right)\right|}\\ \mathbf{if}\;t\_0 \leq 0.0002:\\ \;\;\;\;\sqrt{\left|\left(\left(-0.3333333333333333 \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (sqrt (fabs (log (- 1.0 z0))))))
  (if (<= t_0 0.0002)
    (sqrt
     (fabs (- (* (* (- (* -0.3333333333333333 z0) 0.5) z0) z0) z0)))
    t_0)))

C

double code(double z0) {
	double t_0 = sqrt(fabs(log((1.0 - z0))));
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = sqrt(fabs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	} else {
		tmp = t_0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(abs(log((1.0d0 - z0))))
    if (t_0 <= 0.0002d0) then
        tmp = sqrt(abs(((((((-0.3333333333333333d0) * z0) - 0.5d0) * z0) * z0) - z0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double t_0 = Math.sqrt(Math.abs(Math.log((1.0 - z0))));
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = Math.sqrt(Math.abs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.sqrt(math.fabs(math.log((1.0 - z0))))
	tmp = 0
	if t_0 <= 0.0002:
		tmp = math.sqrt(math.fabs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0)
	t_0 = sqrt(abs(log(Float64(1.0 - z0))))
	tmp = 0.0
	if (t_0 <= 0.0002)
		tmp = sqrt(abs(Float64(Float64(Float64(Float64(Float64(-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = sqrt(abs(log((1.0 - z0))));
	tmp = 0.0;
	if (t_0 <= 0.0002)
		tmp = sqrt(abs((((((-0.3333333333333333 * z0) - 0.5) * z0) * z0) - z0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[Log[N[(1.0 - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0002], N[Sqrt[N[Abs[N[(N[(N[(N[(N[(-0.3333333333333333 * z0), $MachinePrecision] - 0.5), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (fabs.f64 (log.f64 (-.f64 #s(literal 1 binary64) z0)))) < 2.0000000000000001e-4

   1. Initial program 37.1%

      \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)}\right|} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)}\right|} \]

      2. lower--.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)\right|} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)\right|} \]

      5. lower-*.f64 68.5%

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(-0.3333333333333333 \cdot z0 - 0.5\right) - 1\right)\right|} \]

   4. Applied rewrites 68.5%

      \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(-0.3333333333333333 \cdot z0 - 0.5\right) - 1\right)}\right|} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - 1\right)}\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

      3. sub-flip N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right|} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) + -1\right)\right|} \]

      5. distribute-rgt-in N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 + \color{blue}{-1 \cdot z0}\right|} \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 + \left(\mathsf{neg}\left(z0\right)\right)\right|} \]

      7. sub-flip-reverse N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

      8. lower--.f64 N/A

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

      9. lower-*.f64 68.5%

         \[\leadsto \sqrt{\left|\left(z0 \cdot \left(-0.3333333333333333 \cdot z0 - 0.5\right)\right) \cdot z0 - z0\right|} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left|\left(z0 \cdot \left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right)\right) \cdot z0 - z0\right|} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left|\left(\left(\frac{-1}{3} \cdot z0 - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

      12. lower-*.f64 68.5%

          \[\leadsto \sqrt{\left|\left(\left(-0.3333333333333333 \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   6. Applied rewrites 68.5%

      \[\leadsto \sqrt{\left|\left(\left(-0.3333333333333333 \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - \color{blue}{z0}\right|} \]

   if 2.0000000000000001e-4 < (sqrt.f64 (fabs.f64 (log.f64 (-.f64 #s(literal 1 binary64) z0))))

   1. Initial program 37.1%

      \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.5% accurate, 2.8× speedup

Math

\[\sqrt{\left|\left(\left(\left(-0.25 \cdot z0 - 0.3333333333333333\right) \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt
 (fabs
  (-
   (* (* (- (* (- (* -0.25 z0) 0.3333333333333333) z0) 0.5) z0) z0)
   z0))))

C

double code(double z0) {
	return sqrt(fabs((((((((-0.25 * z0) - 0.3333333333333333) * z0) - 0.5) * z0) * z0) - z0)));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs(((((((((-0.25d0) * z0) - 0.3333333333333333d0) * z0) - 0.5d0) * z0) * z0) - z0)))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs((((((((-0.25 * z0) - 0.3333333333333333) * z0) - 0.5) * z0) * z0) - z0)));
}

Python

def code(z0):
	return math.sqrt(math.fabs((((((((-0.25 * z0) - 0.3333333333333333) * z0) - 0.5) * z0) * z0) - z0)))

Julia

function code(z0)
	return sqrt(abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * z0) - 0.3333333333333333) * z0) - 0.5) * z0) * z0) - z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs((((((((-0.25 * z0) - 0.3333333333333333) * z0) - 0.5) * z0) * z0) - z0)));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[(N[(N[(N[(N[(N[(N[(-0.25 * z0), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision] - 0.5), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]

   2. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   4. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   6. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   7. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right|} \]

4. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right|} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

   3. sub-flip N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right|} \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right|} \]

   5. distribute-rgt-in N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 + \color{blue}{-1 \cdot z0}\right|} \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 + \left(\mathsf{neg}\left(z0\right)\right)\right|} \]

   7. sub-flip-reverse N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

   8. lower--.f64 N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

   9. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right)\right) \cdot z0 - z0\right|} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 - z0\right|} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\left|\left(\left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   12. lower-*.f64 68.5%

       \[\leadsto \sqrt{\left|\left(\left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left|\left(\left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   14. *-commutative N/A

       \[\leadsto \sqrt{\left|\left(\left(\left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) \cdot z0 - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   15. lower-*.f64 68.5%

       \[\leadsto \sqrt{\left|\left(\left(\left(-0.25 \cdot z0 - 0.3333333333333333\right) \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

6. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\left(\left(\left(-0.25 \cdot z0 - 0.3333333333333333\right) \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - \color{blue}{z0}\right|} \]
7. Add Preprocessing

Alternative 4: 68.5% accurate, 2.8× speedup

Math

\[\sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt
 (fabs
  (*
   z0
   (-
    (* z0 (- (* z0 (- (* -0.25 z0) 0.3333333333333333)) 0.5))
    1.0)))))

C

double code(double z0) {
	return sqrt(fabs((z0 * ((z0 * ((z0 * ((-0.25 * z0) - 0.3333333333333333)) - 0.5)) - 1.0))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs((z0 * ((z0 * ((z0 * (((-0.25d0) * z0) - 0.3333333333333333d0)) - 0.5d0)) - 1.0d0))))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs((z0 * ((z0 * ((z0 * ((-0.25 * z0) - 0.3333333333333333)) - 0.5)) - 1.0))));
}

Python

def code(z0):
	return math.sqrt(math.fabs((z0 * ((z0 * ((z0 * ((-0.25 * z0) - 0.3333333333333333)) - 0.5)) - 1.0))))

Julia

function code(z0)
	return sqrt(abs(Float64(z0 * Float64(Float64(z0 * Float64(Float64(z0 * Float64(Float64(-0.25 * z0) - 0.3333333333333333)) - 0.5)) - 1.0))))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs((z0 * ((z0 * ((z0 * ((-0.25 * z0) - 0.3333333333333333)) - 0.5)) - 1.0))));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[(z0 * N[(N[(z0 * N[(N[(z0 * N[(N[(-0.25 * z0), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]

   2. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   4. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   6. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   7. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right|} \]

4. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right|} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 4.0× speedup

Math

\[\sqrt{\left|z0 \cdot \left(\left(-0.5 \cdot z0 - 0.5\right) - 0.5\right)\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt (fabs (* z0 (- (- (* -0.5 z0) 0.5) 0.5)))))

C

double code(double z0) {
	return sqrt(fabs((z0 * (((-0.5 * z0) - 0.5) - 0.5))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs((z0 * ((((-0.5d0) * z0) - 0.5d0) - 0.5d0))))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs((z0 * (((-0.5 * z0) - 0.5) - 0.5))));
}

Python

def code(z0):
	return math.sqrt(math.fabs((z0 * (((-0.5 * z0) - 0.5) - 0.5))))

Julia

function code(z0)
	return sqrt(abs(Float64(z0 * Float64(Float64(Float64(-0.5 * z0) - 0.5) - 0.5))))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs((z0 * (((-0.5 * z0) - 0.5) - 0.5))));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[(z0 * N[(N[(N[(-0.5 * z0), $MachinePrecision] - 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]

   2. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   4. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   6. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   7. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right|} \]

4. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right|} \]

5. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \frac{-1}{2} - 1\right)\right|} \]
6. Step-by-step derivation

7. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot -0.5 - 1\right)\right|} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate--r+ N/A

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

   4. lower--.f64 N/A

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

   5. lower--.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(\left(z0 \cdot -0.5 - 0.5\right) - 0.5\right)\right|} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(\left(z0 \cdot \frac{-1}{2} - \frac{1}{2}\right) - \frac{1}{2}\right)\right|} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(\left(\frac{-1}{2} \cdot z0 - \frac{1}{2}\right) - \frac{1}{2}\right)\right|} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(\left(\frac{-1}{2} \cdot z0 - \frac{1}{2}\right) - \frac{1}{2}\right)\right|} \]

   9. lift-*.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(\left(-0.5 \cdot z0 - 0.5\right) - 0.5\right)\right|} \]

   10. *-commutative 68.5%

       \[\leadsto \sqrt{\left|z0 \cdot \left(\left(-0.5 \cdot z0 - 0.5\right) - 0.5\right)\right|} \]

9. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|z0 \cdot \left(\left(-0.5 \cdot z0 - 0.5\right) - \color{blue}{0.5}\right)\right|} \]
10. Add Preprocessing

Alternative 6: 68.5% accurate, 4.5× speedup

Math

\[\sqrt{\left|\left(-0.5 \cdot z0\right) \cdot z0 - z0\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt (fabs (- (* (* -0.5 z0) z0) z0))))

C

double code(double z0) {
	return sqrt(fabs((((-0.5 * z0) * z0) - z0)));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs(((((-0.5d0) * z0) * z0) - z0)))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs((((-0.5 * z0) * z0) - z0)));
}

Python

def code(z0):
	return math.sqrt(math.fabs((((-0.5 * z0) * z0) - z0)))

Julia

function code(z0)
	return sqrt(abs(Float64(Float64(Float64(-0.5 * z0) * z0) - z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs((((-0.5 * z0) * z0) - z0)));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[(N[(N[(-0.5 * z0), $MachinePrecision] * z0), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]

   2. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   4. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   6. lower--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right|} \]

   7. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right|} \]

4. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right|} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \color{blue}{\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right|} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right)\right|} \]

   3. sub-flip N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right|} \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{\left|z0 \cdot \left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right|} \]

   5. distribute-rgt-in N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 + \color{blue}{-1 \cdot z0}\right|} \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 + \left(\mathsf{neg}\left(z0\right)\right)\right|} \]

   7. sub-flip-reverse N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

   8. lower--.f64 N/A

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 - \color{blue}{z0}\right|} \]

   9. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right)\right) \cdot z0 - z0\right|} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left|\left(z0 \cdot \left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right) \cdot z0 - z0\right|} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\left|\left(\left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   12. lower-*.f64 68.5%

       \[\leadsto \sqrt{\left|\left(\left(z0 \cdot \left(-0.25 \cdot z0 - 0.3333333333333333\right) - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left|\left(\left(z0 \cdot \left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   14. *-commutative N/A

       \[\leadsto \sqrt{\left|\left(\left(\left(\frac{-1}{4} \cdot z0 - \frac{1}{3}\right) \cdot z0 - \frac{1}{2}\right) \cdot z0\right) \cdot z0 - z0\right|} \]

   15. lower-*.f64 68.5%

       \[\leadsto \sqrt{\left|\left(\left(\left(-0.25 \cdot z0 - 0.3333333333333333\right) \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - z0\right|} \]

6. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\left(\left(\left(-0.25 \cdot z0 - 0.3333333333333333\right) \cdot z0 - 0.5\right) \cdot z0\right) \cdot z0 - \color{blue}{z0}\right|} \]

7. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\left(\frac{-1}{2} \cdot z0\right) \cdot z0 - z0\right|} \]
8. Step-by-step derivation

9. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\left(-0.5 \cdot z0\right) \cdot z0 - z0\right|} \]
10. Add Preprocessing

Alternative 7: 68.5% accurate, 4.5× speedup

Math

\[\sqrt{\left|z0 \cdot \left(-0.5 \cdot z0 - 1\right)\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt (fabs (* z0 (- (* -0.5 z0) 1.0)))))

C

double code(double z0) {
	return sqrt(fabs((z0 * ((-0.5 * z0) - 1.0))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs((z0 * (((-0.5d0) * z0) - 1.0d0))))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs((z0 * ((-0.5 * z0) - 1.0))));
}

Python

def code(z0):
	return math.sqrt(math.fabs((z0 * ((-0.5 * z0) - 1.0))))

Julia

function code(z0)
	return sqrt(abs(Float64(z0 * Float64(Float64(-0.5 * z0) - 1.0))))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs((z0 * ((-0.5 * z0) - 1.0))));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[N[(z0 * N[(N[(-0.5 * z0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(\frac{-1}{2} \cdot z0 - 1\right)}\right|} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|z0 \cdot \left(-0.5 \cdot z0 - 1\right)\right|} \]

4. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\color{blue}{z0 \cdot \left(-0.5 \cdot z0 - 1\right)}\right|} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 7.7× speedup

Math

\[\sqrt{\left|-z0\right|} \]

FPCore

(FPCore (z0)
  :precision binary64
  (sqrt (fabs (- z0))))

C

double code(double z0) {
	return sqrt(fabs(-z0));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sqrt(abs(-z0))
end function

Java

public static double code(double z0) {
	return Math.sqrt(Math.abs(-z0));
}

Python

def code(z0):
	return math.sqrt(math.fabs(-z0))

Julia

function code(z0)
	return sqrt(abs(Float64(-z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sqrt(abs(-z0));
end

Wolfram

code[z0_] := N[Sqrt[N[Abs[(-z0)], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\left|\log \left(1 - z0\right)\right|} \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\left|\color{blue}{-1 \cdot z0}\right|} \]
3. Step-by-step derivation

   1. lower-*.f64 68.5%

      \[\leadsto \sqrt{\left|-1 \cdot \color{blue}{z0}\right|} \]

4. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|\color{blue}{-1 \cdot z0}\right|} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left|-1 \cdot \color{blue}{z0}\right|} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt{\left|\mathsf{neg}\left(z0\right)\right|} \]

   3. lower-neg.f64 68.5%

      \[\leadsto \sqrt{\left|-z0\right|} \]

6. Applied rewrites 68.5%

   \[\leadsto \sqrt{\left|-z0\right|} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(sqrt (fabs (log (- 1 z0))))"
  :precision binary64
  (sqrt (fabs (log (- 1.0 z0)))))