(/ -1 (+ (/ z2 (* z3 z3)) (/ z0 (* z1 z1))))

Percentage Accurate: 81.3% → 96.1%

Time: 2.8s

Alternatives: 9

Speedup: 0.8×

Specification

Math

\[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (/ -1.0 (+ (/ z2 (* z3 z3)) (/ z0 (* z1 z1)))))

C

double code(double z2, double z3, double z0, double z1) {
	return -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * 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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = (-1.0d0) / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)))
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	return -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)));
}

Python

def code(z2, z3, z0, z1):
	return -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)))

Julia

function code(z2, z3, z0, z1)
	return Float64(-1.0 / Float64(Float64(z2 / Float64(z3 * z3)) + Float64(z0 / Float64(z1 * z1))))
end

MATLAB

function tmp = code(z2, z3, z0, z1)
	tmp = -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)));
end

Wolfram

code[z2_, z3_, z0_, z1_] := N[(-1.0 / N[(N[(z2 / N[(z3 * z3), $MachinePrecision]), $MachinePrecision] + N[(z0 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 81.3% accurate, 1.0× speedup

Math

\[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (/ -1.0 (+ (/ z2 (* z3 z3)) (/ z0 (* z1 z1)))))

C

double code(double z2, double z3, double z0, double z1) {
	return -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * 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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = (-1.0d0) / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)))
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	return -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)));
}

Python

def code(z2, z3, z0, z1):
	return -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)))

Julia

function code(z2, z3, z0, z1)
	return Float64(-1.0 / Float64(Float64(z2 / Float64(z3 * z3)) + Float64(z0 / Float64(z1 * z1))))
end

MATLAB

function tmp = code(z2, z3, z0, z1)
	tmp = -1.0 / ((z2 / (z3 * z3)) + (z0 / (z1 * z1)));
end

Wolfram

code[z2_, z3_, z0_, z1_] := N[(-1.0 / N[(N[(z2 / N[(z3 * z3), $MachinePrecision]), $MachinePrecision] + N[(z0 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z1 \cdot z1 \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\left(\frac{-1}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3} \cdot z3\right) \cdot z1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (if (<= (* z1 z1) 5e-251)
  (* (* (/ -1.0 (+ (* z1 (/ z2 z3)) (* (/ z0 z1) z3))) z3) z1)
  (/ -1.0 (+ (/ (/ z2 z3) z3) (/ (/ z0 z1) z1)))))

C

double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if ((z1 * z1) <= 5e-251) {
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1;
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + ((z0 / z1) / 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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: tmp
    if ((z1 * z1) <= 5d-251) then
        tmp = (((-1.0d0) / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1
    else
        tmp = (-1.0d0) / (((z2 / z3) / z3) + ((z0 / z1) / z1))
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if ((z1 * z1) <= 5e-251) {
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1;
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + ((z0 / z1) / z1));
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	tmp = 0
	if (z1 * z1) <= 5e-251:
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1
	else:
		tmp = -1.0 / (((z2 / z3) / z3) + ((z0 / z1) / z1))
	return tmp

Julia

function code(z2, z3, z0, z1)
	tmp = 0.0
	if (Float64(z1 * z1) <= 5e-251)
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(z1 * Float64(z2 / z3)) + Float64(Float64(z0 / z1) * z3))) * z3) * z1);
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(z2 / z3) / z3) + Float64(Float64(z0 / z1) / z1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	tmp = 0.0;
	if ((z1 * z1) <= 5e-251)
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1;
	else
		tmp = -1.0 / (((z2 / z3) / z3) + ((z0 / z1) / z1));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := If[LessEqual[N[(z1 * z1), $MachinePrecision], 5e-251], N[(N[(N[(-1.0 / N[(N[(z1 * N[(z2 / z3), $MachinePrecision]), $MachinePrecision] + N[(N[(z0 / z1), $MachinePrecision] * z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z3), $MachinePrecision] * z1), $MachinePrecision], N[(-1.0 / N[(N[(N[(z2 / z3), $MachinePrecision] / z3), $MachinePrecision] + N[(N[(z0 / z1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z1 z1) < 5.0000000000000003e-251

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\color{blue}{z1 \cdot z1}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. lower-/.f64 94.5%

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\color{blue}{\frac{z0}{z1}}}{z1}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{\frac{z0}{z1}}{z1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}{z3 \cdot z1}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}} \cdot \left(z3 \cdot z1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3} \cdot z1} + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3} + \frac{z2}{z3} \cdot z1} \cdot \left(z3 \cdot z1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{\frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

      17. lower-*.f64 84.4%

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

   7. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z3 \cdot z1\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot z3\right) \cdot z1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot z3\right) \cdot z1} \]

      6. lower-*.f64 87.1%

         \[\leadsto \color{blue}{\left(\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot z3\right)} \cdot z1 \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}}} \cdot z3\right) \cdot z1 \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{-1}{\color{blue}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3}} \cdot z3\right) \cdot z1 \]

      9. lower-+.f64 87.1%

         \[\leadsto \left(\frac{-1}{\color{blue}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3}} \cdot z3\right) \cdot z1 \]

   9. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\left(\frac{-1}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3} \cdot z3\right) \cdot z1} \]

   if 5.0000000000000003e-251 < (*.f64 z1 z1)

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\color{blue}{z1 \cdot z1}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. lower-/.f64 94.5%

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\color{blue}{\frac{z0}{z1}}}{z1}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z1 \cdot z1 \leq 5 \cdot 10^{-251}:\\ \;\;\;\;\left(\frac{-1}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3} \cdot z3\right) \cdot z1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{z1 \cdot z1}}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (if (<= (* z1 z1) 5e-251)
  (* (* (/ -1.0 (+ (* z1 (/ z2 z3)) (* (/ z0 z1) z3))) z3) z1)
  (/ -1.0 (+ (/ (/ z2 z3) z3) (/ z0 (* z1 z1))))))

C

double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if ((z1 * z1) <= 5e-251) {
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1;
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (z1 * 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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: tmp
    if ((z1 * z1) <= 5d-251) then
        tmp = (((-1.0d0) / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1
    else
        tmp = (-1.0d0) / (((z2 / z3) / z3) + (z0 / (z1 * z1)))
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if ((z1 * z1) <= 5e-251) {
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1;
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (z1 * z1)));
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	tmp = 0
	if (z1 * z1) <= 5e-251:
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1
	else:
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (z1 * z1)))
	return tmp

Julia

function code(z2, z3, z0, z1)
	tmp = 0.0
	if (Float64(z1 * z1) <= 5e-251)
		tmp = Float64(Float64(Float64(-1.0 / Float64(Float64(z1 * Float64(z2 / z3)) + Float64(Float64(z0 / z1) * z3))) * z3) * z1);
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(z2 / z3) / z3) + Float64(z0 / Float64(z1 * z1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	tmp = 0.0;
	if ((z1 * z1) <= 5e-251)
		tmp = ((-1.0 / ((z1 * (z2 / z3)) + ((z0 / z1) * z3))) * z3) * z1;
	else
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (z1 * z1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := If[LessEqual[N[(z1 * z1), $MachinePrecision], 5e-251], N[(N[(N[(-1.0 / N[(N[(z1 * N[(z2 / z3), $MachinePrecision]), $MachinePrecision] + N[(N[(z0 / z1), $MachinePrecision] * z3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z3), $MachinePrecision] * z1), $MachinePrecision], N[(-1.0 / N[(N[(N[(z2 / z3), $MachinePrecision] / z3), $MachinePrecision] + N[(z0 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z1 z1) < 5.0000000000000003e-251

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\color{blue}{z1 \cdot z1}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. lower-/.f64 94.5%

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\color{blue}{\frac{z0}{z1}}}{z1}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{\frac{z0}{z1}}{z1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}{z3 \cdot z1}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}} \cdot \left(z3 \cdot z1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3} \cdot z1} + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3} + \frac{z2}{z3} \cdot z1} \cdot \left(z3 \cdot z1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{\frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

      17. lower-*.f64 84.4%

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

   7. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z3 \cdot z1\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot z3\right) \cdot z1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot z3\right) \cdot z1} \]

      6. lower-*.f64 87.1%

         \[\leadsto \color{blue}{\left(\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot z3\right)} \cdot z1 \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}}} \cdot z3\right) \cdot z1 \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{-1}{\color{blue}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3}} \cdot z3\right) \cdot z1 \]

      9. lower-+.f64 87.1%

         \[\leadsto \left(\frac{-1}{\color{blue}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3}} \cdot z3\right) \cdot z1 \]

   9. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\left(\frac{-1}{z1 \cdot \frac{z2}{z3} + \frac{z0}{z1} \cdot z3} \cdot z3\right) \cdot z1} \]

   if 5.0000000000000003e-251 < (*.f64 z1 z1)

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 10^{-125}:\\ \;\;\;\;\frac{\left|z1\right| \cdot z3}{-\left(\left|z1\right| \cdot \frac{z2}{z3} + \frac{z0}{\left|z1\right|} \cdot z3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\left|z1\right| \cdot \left|z1\right|}}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (if (<= (fabs z1) 1e-125)
  (/
   (* (fabs z1) z3)
   (- (+ (* (fabs z1) (/ z2 z3)) (* (/ z0 (fabs z1)) z3))))
  (/ -1.0 (+ (/ (/ z2 z3) z3) (/ z0 (* (fabs z1) (fabs z1)))))))

C

double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (fabs(z1) <= 1e-125) {
		tmp = (fabs(z1) * z3) / -((fabs(z1) * (z2 / z3)) + ((z0 / fabs(z1)) * z3));
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (fabs(z1) * fabs(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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: tmp
    if (abs(z1) <= 1d-125) then
        tmp = (abs(z1) * z3) / -((abs(z1) * (z2 / z3)) + ((z0 / abs(z1)) * z3))
    else
        tmp = (-1.0d0) / (((z2 / z3) / z3) + (z0 / (abs(z1) * abs(z1))))
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (Math.abs(z1) <= 1e-125) {
		tmp = (Math.abs(z1) * z3) / -((Math.abs(z1) * (z2 / z3)) + ((z0 / Math.abs(z1)) * z3));
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (Math.abs(z1) * Math.abs(z1))));
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	tmp = 0
	if math.fabs(z1) <= 1e-125:
		tmp = (math.fabs(z1) * z3) / -((math.fabs(z1) * (z2 / z3)) + ((z0 / math.fabs(z1)) * z3))
	else:
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (math.fabs(z1) * math.fabs(z1))))
	return tmp

Julia

function code(z2, z3, z0, z1)
	tmp = 0.0
	if (abs(z1) <= 1e-125)
		tmp = Float64(Float64(abs(z1) * z3) / Float64(-Float64(Float64(abs(z1) * Float64(z2 / z3)) + Float64(Float64(z0 / abs(z1)) * z3))));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(z2 / z3) / z3) + Float64(z0 / Float64(abs(z1) * abs(z1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	tmp = 0.0;
	if (abs(z1) <= 1e-125)
		tmp = (abs(z1) * z3) / -((abs(z1) * (z2 / z3)) + ((z0 / abs(z1)) * z3));
	else
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (abs(z1) * abs(z1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := If[LessEqual[N[Abs[z1], $MachinePrecision], 1e-125], N[(N[(N[Abs[z1], $MachinePrecision] * z3), $MachinePrecision] / (-N[(N[(N[Abs[z1], $MachinePrecision] * N[(z2 / z3), $MachinePrecision]), $MachinePrecision] + N[(N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision] * z3), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[(N[(z2 / z3), $MachinePrecision] / z3), $MachinePrecision] + N[(z0 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z1 < 1e-125

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\color{blue}{z1 \cdot z1}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. lower-/.f64 94.5%

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\color{blue}{\frac{z0}{z1}}}{z1}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{\frac{z0}{z1}}{z1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}{z3 \cdot z1}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}} \cdot \left(z3 \cdot z1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3} \cdot z1} + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3} + \frac{z2}{z3} \cdot z1} \cdot \left(z3 \cdot z1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{\frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

      17. lower-*.f64 84.4%

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

   7. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z1 \cdot z3\right) \cdot \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(z1 \cdot z3\right) \cdot \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}}} \]

      4. frac-2neg N/A

         \[\leadsto \left(z1 \cdot z3\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \left(z1 \cdot z3\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}\right)\right)} \]

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 84.5%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 84.5%

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

   9. Applied rewrites 84.5%

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

   if 1e-125 < z1

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 4 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left|z1\right|}{z0} \cdot \left(-\left|z1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\left|z1\right| \cdot \left|z1\right|}}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (if (<= (fabs z1) 4e-149)
  (* (/ (fabs z1) z0) (- (fabs z1)))
  (/ -1.0 (+ (/ (/ z2 z3) z3) (/ z0 (* (fabs z1) (fabs z1)))))))

C

double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (fabs(z1) <= 4e-149) {
		tmp = (fabs(z1) / z0) * -fabs(z1);
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (fabs(z1) * fabs(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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: tmp
    if (abs(z1) <= 4d-149) then
        tmp = (abs(z1) / z0) * -abs(z1)
    else
        tmp = (-1.0d0) / (((z2 / z3) / z3) + (z0 / (abs(z1) * abs(z1))))
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (Math.abs(z1) <= 4e-149) {
		tmp = (Math.abs(z1) / z0) * -Math.abs(z1);
	} else {
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (Math.abs(z1) * Math.abs(z1))));
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	tmp = 0
	if math.fabs(z1) <= 4e-149:
		tmp = (math.fabs(z1) / z0) * -math.fabs(z1)
	else:
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (math.fabs(z1) * math.fabs(z1))))
	return tmp

Julia

function code(z2, z3, z0, z1)
	tmp = 0.0
	if (abs(z1) <= 4e-149)
		tmp = Float64(Float64(abs(z1) / z0) * Float64(-abs(z1)));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(z2 / z3) / z3) + Float64(z0 / Float64(abs(z1) * abs(z1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	tmp = 0.0;
	if (abs(z1) <= 4e-149)
		tmp = (abs(z1) / z0) * -abs(z1);
	else
		tmp = -1.0 / (((z2 / z3) / z3) + (z0 / (abs(z1) * abs(z1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := If[LessEqual[N[Abs[z1], $MachinePrecision], 4e-149], N[(N[(N[Abs[z1], $MachinePrecision] / z0), $MachinePrecision] * (-N[Abs[z1], $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[(N[(z2 / z3), $MachinePrecision] / z3), $MachinePrecision] + N[(z0 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z1 < 3.9999999999999999e-149

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}\right)\right)} \]

      6. add-to-fraction N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{z1 \cdot z1}}\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{\mathsf{neg}\left(z1 \cdot z1\right)}}} \]

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.5%

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z2 around 0

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.8%

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-z1\right) \cdot z1}{z0}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-z1\right) \cdot z1}}{z0} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-z1\right) \cdot \frac{z1}{z0}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      6. lower-/.f64 55.0%

         \[\leadsto \color{blue}{\frac{z1}{z0}} \cdot \left(-z1\right) \]

   8. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

   if 3.9999999999999999e-149 < z1

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{z2}{z3 \cdot z3}\\ \mathbf{if}\;t\_0 + \frac{z0}{z1 \cdot z1} \leq \infty:\\ \;\;\;\;\frac{-1}{t\_0 + \frac{\frac{z0}{z1}}{z1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z1 \cdot z1\right) \cdot z3}{-z0 \cdot z3}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (let* ((t_0 (/ z2 (* z3 z3))))
  (if (<= (+ t_0 (/ z0 (* z1 z1))) INFINITY)
    (/ -1.0 (+ t_0 (/ (/ z0 z1) z1)))
    (/ (* (* z1 z1) z3) (- (* z0 z3))))))

C

double code(double z2, double z3, double z0, double z1) {
	double t_0 = z2 / (z3 * z3);
	double tmp;
	if ((t_0 + (z0 / (z1 * z1))) <= ((double) INFINITY)) {
		tmp = -1.0 / (t_0 + ((z0 / z1) / z1));
	} else {
		tmp = ((z1 * z1) * z3) / -(z0 * z3);
	}
	return tmp;
}

Java

public static double code(double z2, double z3, double z0, double z1) {
	double t_0 = z2 / (z3 * z3);
	double tmp;
	if ((t_0 + (z0 / (z1 * z1))) <= Double.POSITIVE_INFINITY) {
		tmp = -1.0 / (t_0 + ((z0 / z1) / z1));
	} else {
		tmp = ((z1 * z1) * z3) / -(z0 * z3);
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	t_0 = z2 / (z3 * z3)
	tmp = 0
	if (t_0 + (z0 / (z1 * z1))) <= math.inf:
		tmp = -1.0 / (t_0 + ((z0 / z1) / z1))
	else:
		tmp = ((z1 * z1) * z3) / -(z0 * z3)
	return tmp

Julia

function code(z2, z3, z0, z1)
	t_0 = Float64(z2 / Float64(z3 * z3))
	tmp = 0.0
	if (Float64(t_0 + Float64(z0 / Float64(z1 * z1))) <= Inf)
		tmp = Float64(-1.0 / Float64(t_0 + Float64(Float64(z0 / z1) / z1)));
	else
		tmp = Float64(Float64(Float64(z1 * z1) * z3) / Float64(-Float64(z0 * z3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	t_0 = z2 / (z3 * z3);
	tmp = 0.0;
	if ((t_0 + (z0 / (z1 * z1))) <= Inf)
		tmp = -1.0 / (t_0 + ((z0 / z1) / z1));
	else
		tmp = ((z1 * z1) * z3) / -(z0 * z3);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := Block[{t$95$0 = N[(z2 / N[(z3 * z3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(z0 / N[(z1 * z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(-1.0 / N[(t$95$0 + N[(N[(z0 / z1), $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z1 * z1), $MachinePrecision] * z3), $MachinePrecision] / (-N[(z0 * z3), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 z2 (*.f64 z3 z3)) (/.f64 z0 (*.f64 z1 z1))) < +inf.0

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{\color{blue}{z1 \cdot z1}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. lower-/.f64 88.0%

         \[\leadsto \frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{\color{blue}{\frac{z0}{z1}}}{z1}} \]

   3. Applied rewrites 88.0%

      \[\leadsto \frac{-1}{\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

   if +inf.0 < (+.f64 (/.f64 z2 (*.f64 z3 z3)) (/.f64 z0 (*.f64 z1 z1)))

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{z1 \cdot z1}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{z1 \cdot z1}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3} \cdot \left(z1 \cdot z1\right) + z0 \cdot z3}{z3 \cdot \left(z1 \cdot z1\right)}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot \left(z1 \cdot z1\right) + z0 \cdot z3} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot \left(z1 \cdot z1\right) + z0 \cdot z3} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot \left(z1 \cdot z1\right) + z0 \cdot z3}} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\color{blue}{z0 \cdot z3 + \frac{z2}{z3} \cdot \left(z1 \cdot z1\right)}} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{z0 \cdot z3 + \frac{z2}{z3} \cdot \left(z1 \cdot z1\right)}} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{z0 \cdot z3} + \frac{z2}{z3} \cdot \left(z1 \cdot z1\right)} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{-1}{z0 \cdot z3 + \frac{z2}{z3} \cdot \color{blue}{\left(z1 \cdot z1\right)}} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{-1}{z0 \cdot z3 + \color{blue}{\left(\frac{z2}{z3} \cdot z1\right) \cdot z1}} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-1}{z0 \cdot z3 + \color{blue}{\left(\frac{z2}{z3} \cdot z1\right) \cdot z1}} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{z0 \cdot z3 + \color{blue}{\left(\frac{z2}{z3} \cdot z1\right)} \cdot z1} \cdot \left(z3 \cdot \left(z1 \cdot z1\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{z0 \cdot z3 + \left(\frac{z2}{z3} \cdot z1\right) \cdot z1} \cdot \color{blue}{\left(\left(z1 \cdot z1\right) \cdot z3\right)} \]

      17. lower-*.f64 63.0%

          \[\leadsto \frac{-1}{z0 \cdot z3 + \left(\frac{z2}{z3} \cdot z1\right) \cdot z1} \cdot \color{blue}{\left(\left(z1 \cdot z1\right) \cdot z3\right)} \]

   5. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\frac{-1}{z0 \cdot z3 + \left(\frac{z2}{z3} \cdot z1\right) \cdot z1} \cdot \left(\left(z1 \cdot z1\right) \cdot z3\right)} \]

   6. Taylor expanded in z2 around 0

      \[\leadsto \frac{-1}{\color{blue}{z0 \cdot z3}} \cdot \left(\left(z1 \cdot z1\right) \cdot z3\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 42.6%

         \[\leadsto \frac{-1}{z0 \cdot \color{blue}{z3}} \cdot \left(\left(z1 \cdot z1\right) \cdot z3\right) \]

   8. Applied rewrites 42.6%

      \[\leadsto \frac{-1}{\color{blue}{z0 \cdot z3}} \cdot \left(\left(z1 \cdot z1\right) \cdot z3\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{z0 \cdot z3} \cdot \left(\left(z1 \cdot z1\right) \cdot z3\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(z1 \cdot z1\right) \cdot z3\right) \cdot \frac{-1}{z0 \cdot z3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(z1 \cdot z1\right) \cdot z3\right) \cdot \color{blue}{\frac{-1}{z0 \cdot z3}} \]

      4. frac-2neg N/A

         \[\leadsto \left(\left(z1 \cdot z1\right) \cdot z3\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(z0 \cdot z3\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(z1 \cdot z1\right) \cdot z3\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z0 \cdot z3\right)} \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\left(z1 \cdot z1\right) \cdot z3}{\mathsf{neg}\left(z0 \cdot z3\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(z1 \cdot z1\right) \cdot z3}{\mathsf{neg}\left(z0 \cdot z3\right)}} \]

      8. lower-neg.f64 42.9%

         \[\leadsto \frac{\left(z1 \cdot z1\right) \cdot z3}{\color{blue}{-z0 \cdot z3}} \]

   10. Applied rewrites 42.9%

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

Alternative 6: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 5.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{\left|z1\right|}{z0} \cdot \left(-\left|z1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z2}{\left(-z3\right) \cdot z3} - \frac{z0}{\left|z1\right| \cdot \left|z1\right|}}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (if (<= (fabs z1) 5.5e-123)
  (* (/ (fabs z1) z0) (- (fabs z1)))
  (/ 1.0 (- (/ z2 (* (- z3) z3)) (/ z0 (* (fabs z1) (fabs z1)))))))

C

double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (fabs(z1) <= 5.5e-123) {
		tmp = (fabs(z1) / z0) * -fabs(z1);
	} else {
		tmp = 1.0 / ((z2 / (-z3 * z3)) - (z0 / (fabs(z1) * fabs(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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: tmp
    if (abs(z1) <= 5.5d-123) then
        tmp = (abs(z1) / z0) * -abs(z1)
    else
        tmp = 1.0d0 / ((z2 / (-z3 * z3)) - (z0 / (abs(z1) * abs(z1))))
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (Math.abs(z1) <= 5.5e-123) {
		tmp = (Math.abs(z1) / z0) * -Math.abs(z1);
	} else {
		tmp = 1.0 / ((z2 / (-z3 * z3)) - (z0 / (Math.abs(z1) * Math.abs(z1))));
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	tmp = 0
	if math.fabs(z1) <= 5.5e-123:
		tmp = (math.fabs(z1) / z0) * -math.fabs(z1)
	else:
		tmp = 1.0 / ((z2 / (-z3 * z3)) - (z0 / (math.fabs(z1) * math.fabs(z1))))
	return tmp

Julia

function code(z2, z3, z0, z1)
	tmp = 0.0
	if (abs(z1) <= 5.5e-123)
		tmp = Float64(Float64(abs(z1) / z0) * Float64(-abs(z1)));
	else
		tmp = Float64(1.0 / Float64(Float64(z2 / Float64(Float64(-z3) * z3)) - Float64(z0 / Float64(abs(z1) * abs(z1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	tmp = 0.0;
	if (abs(z1) <= 5.5e-123)
		tmp = (abs(z1) / z0) * -abs(z1);
	else
		tmp = 1.0 / ((z2 / (-z3 * z3)) - (z0 / (abs(z1) * abs(z1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := If[LessEqual[N[Abs[z1], $MachinePrecision], 5.5e-123], N[(N[(N[Abs[z1], $MachinePrecision] / z0), $MachinePrecision] * (-N[Abs[z1], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[(N[(z2 / N[((-z3) * z3), $MachinePrecision]), $MachinePrecision] - N[(z0 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z1 < 5.5e-123

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}\right)\right)} \]

      6. add-to-fraction N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{z1 \cdot z1}}\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{\mathsf{neg}\left(z1 \cdot z1\right)}}} \]

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.5%

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z2 around 0

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.8%

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-z1\right) \cdot z1}{z0}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-z1\right) \cdot z1}}{z0} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-z1\right) \cdot \frac{z1}{z0}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      6. lower-/.f64 55.0%

         \[\leadsto \color{blue}{\frac{z1}{z0}} \cdot \left(-z1\right) \]

   8. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

   if 5.5e-123 < z1

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)}\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z0}{z1 \cdot z1} + \frac{z2}{z3 \cdot z3}\right)}\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z0}{z1 \cdot z1} + \color{blue}{\frac{z2}{z3 \cdot z3}}\right)\right)} \]

      8. mult-flip N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z0}{z1 \cdot z1} + \color{blue}{z2 \cdot \frac{1}{z3 \cdot z3}}\right)\right)} \]

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

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z0}{z1 \cdot z1} - \left(\mathsf{neg}\left(z2\right)\right) \cdot \frac{1}{z3 \cdot z3}\right)}\right)} \]

      10. sub-negate N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \frac{1}{z3 \cdot z3} - \frac{z0}{z1 \cdot z1}}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(z2\right)\right) \cdot \frac{1}{z3 \cdot z3} - \frac{z0}{z1 \cdot z1}}} \]

   3. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{z2}{\left(-z3\right) \cdot z3} - \frac{z0}{z1 \cdot z1}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 5.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{\left|z1\right|}{z0} \cdot \left(-\left|z1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{\left|z1\right| \cdot \left|z1\right|}}\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (if (<= (fabs z1) 5.5e-123)
  (* (/ (fabs z1) z0) (- (fabs z1)))
  (/ -1.0 (+ (/ z2 (* z3 z3)) (/ z0 (* (fabs z1) (fabs z1)))))))

C

double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (fabs(z1) <= 5.5e-123) {
		tmp = (fabs(z1) / z0) * -fabs(z1);
	} else {
		tmp = -1.0 / ((z2 / (z3 * z3)) + (z0 / (fabs(z1) * fabs(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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: tmp
    if (abs(z1) <= 5.5d-123) then
        tmp = (abs(z1) / z0) * -abs(z1)
    else
        tmp = (-1.0d0) / ((z2 / (z3 * z3)) + (z0 / (abs(z1) * abs(z1))))
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double tmp;
	if (Math.abs(z1) <= 5.5e-123) {
		tmp = (Math.abs(z1) / z0) * -Math.abs(z1);
	} else {
		tmp = -1.0 / ((z2 / (z3 * z3)) + (z0 / (Math.abs(z1) * Math.abs(z1))));
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	tmp = 0
	if math.fabs(z1) <= 5.5e-123:
		tmp = (math.fabs(z1) / z0) * -math.fabs(z1)
	else:
		tmp = -1.0 / ((z2 / (z3 * z3)) + (z0 / (math.fabs(z1) * math.fabs(z1))))
	return tmp

Julia

function code(z2, z3, z0, z1)
	tmp = 0.0
	if (abs(z1) <= 5.5e-123)
		tmp = Float64(Float64(abs(z1) / z0) * Float64(-abs(z1)));
	else
		tmp = Float64(-1.0 / Float64(Float64(z2 / Float64(z3 * z3)) + Float64(z0 / Float64(abs(z1) * abs(z1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	tmp = 0.0;
	if (abs(z1) <= 5.5e-123)
		tmp = (abs(z1) / z0) * -abs(z1);
	else
		tmp = -1.0 / ((z2 / (z3 * z3)) + (z0 / (abs(z1) * abs(z1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := If[LessEqual[N[Abs[z1], $MachinePrecision], 5.5e-123], N[(N[(N[Abs[z1], $MachinePrecision] / z0), $MachinePrecision] * (-N[Abs[z1], $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[(z2 / N[(z3 * z3), $MachinePrecision]), $MachinePrecision] + N[(z0 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z1 < 5.5e-123

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}\right)\right)} \]

      6. add-to-fraction N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{z1 \cdot z1}}\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{\mathsf{neg}\left(z1 \cdot z1\right)}}} \]

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.5%

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z2 around 0

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.8%

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-z1\right) \cdot z1}{z0}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-z1\right) \cdot z1}}{z0} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-z1\right) \cdot \frac{z1}{z0}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      6. lower-/.f64 55.0%

         \[\leadsto \color{blue}{\frac{z1}{z0}} \cdot \left(-z1\right) \]

   8. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

   if 5.5e-123 < z1

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{z2}{z3 \cdot z3}\\ t_1 := \left(-1 \cdot \frac{z3}{z1 \cdot z2}\right) \cdot \left(z1 \cdot z3\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{z1}{z0} \cdot \left(-z1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (let* ((t_0 (/ z2 (* z3 z3)))
       (t_1 (* (* -1.0 (/ z3 (* z1 z2))) (* z1 z3))))
  (if (<= t_0 -1e+200)
    t_1
    (if (<= t_0 2e-5) (* (/ z1 z0) (- z1)) t_1))))

C

double code(double z2, double z3, double z0, double z1) {
	double t_0 = z2 / (z3 * z3);
	double t_1 = (-1.0 * (z3 / (z1 * z2))) * (z1 * z3);
	double tmp;
	if (t_0 <= -1e+200) {
		tmp = t_1;
	} else if (t_0 <= 2e-5) {
		tmp = (z1 / z0) * -z1;
	} 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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z2 / (z3 * z3)
    t_1 = ((-1.0d0) * (z3 / (z1 * z2))) * (z1 * z3)
    if (t_0 <= (-1d+200)) then
        tmp = t_1
    else if (t_0 <= 2d-5) then
        tmp = (z1 / z0) * -z1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	double t_0 = z2 / (z3 * z3);
	double t_1 = (-1.0 * (z3 / (z1 * z2))) * (z1 * z3);
	double tmp;
	if (t_0 <= -1e+200) {
		tmp = t_1;
	} else if (t_0 <= 2e-5) {
		tmp = (z1 / z0) * -z1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z3, z0, z1):
	t_0 = z2 / (z3 * z3)
	t_1 = (-1.0 * (z3 / (z1 * z2))) * (z1 * z3)
	tmp = 0
	if t_0 <= -1e+200:
		tmp = t_1
	elif t_0 <= 2e-5:
		tmp = (z1 / z0) * -z1
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z3, z0, z1)
	t_0 = Float64(z2 / Float64(z3 * z3))
	t_1 = Float64(Float64(-1.0 * Float64(z3 / Float64(z1 * z2))) * Float64(z1 * z3))
	tmp = 0.0
	if (t_0 <= -1e+200)
		tmp = t_1;
	elseif (t_0 <= 2e-5)
		tmp = Float64(Float64(z1 / z0) * Float64(-z1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z3, z0, z1)
	t_0 = z2 / (z3 * z3);
	t_1 = (-1.0 * (z3 / (z1 * z2))) * (z1 * z3);
	tmp = 0.0;
	if (t_0 <= -1e+200)
		tmp = t_1;
	elseif (t_0 <= 2e-5)
		tmp = (z1 / z0) * -z1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z3_, z0_, z1_] := Block[{t$95$0 = N[(z2 / N[(z3 * z3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 * N[(z3 / N[(z1 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z1 * z3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+200], t$95$1, If[LessEqual[t$95$0, 2e-5], N[(N[(z1 / z0), $MachinePrecision] * (-z1)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z2 (*.f64 z3 z3)) < -9.9999999999999997e199 or 2.0000000000000002e-5 < (/.f64 z2 (*.f64 z3 z3))

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{z2}{\color{blue}{z3 \cdot z3}} + \frac{z0}{z1 \cdot z1}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]

      5. lower-/.f64 87.6%

         \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{z2}{z3}}}{z3} + \frac{z0}{z1 \cdot z1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{z0}{z1 \cdot z1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{z0}{\color{blue}{z1 \cdot z1}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. lower-/.f64 94.5%

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\color{blue}{\frac{z0}{z1}}}{z1}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3} + \frac{\frac{z0}{z1}}{z1}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3}}{z3}} + \frac{\frac{z0}{z1}}{z1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\frac{z2}{z3}}{z3} + \color{blue}{\frac{\frac{z0}{z1}}{z1}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}{z3 \cdot z1}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3} \cdot z1 + \frac{z0}{z1} \cdot z3}} \cdot \left(z3 \cdot z1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{z2}{z3} \cdot z1} + \frac{z0}{z1} \cdot z3} \cdot \left(z3 \cdot z1\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3 + \frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{z0}{z1} \cdot z3} + \frac{z2}{z3} \cdot z1} \cdot \left(z3 \cdot z1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{\frac{z2}{z3} \cdot z1}} \cdot \left(z3 \cdot z1\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + \color{blue}{z1 \cdot \frac{z2}{z3}}} \cdot \left(z3 \cdot z1\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

      17. lower-*.f64 84.4%

          \[\leadsto \frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \color{blue}{\left(z1 \cdot z3\right)} \]

   7. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{z0}{z1} \cdot z3 + z1 \cdot \frac{z2}{z3}} \cdot \left(z1 \cdot z3\right)} \]

   8. Taylor expanded in z2 around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{z3}{z1 \cdot z2}\right)} \cdot \left(z1 \cdot z3\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\frac{z3}{z1 \cdot z2}}\right) \cdot \left(z1 \cdot z3\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{z3}{\color{blue}{z1 \cdot z2}}\right) \cdot \left(z1 \cdot z3\right) \]

      3. lower-*.f64 46.5%

         \[\leadsto \left(-1 \cdot \frac{z3}{z1 \cdot \color{blue}{z2}}\right) \cdot \left(z1 \cdot z3\right) \]

   10. Applied rewrites 46.5%

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{z3}{z1 \cdot z2}\right)} \cdot \left(z1 \cdot z3\right) \]

   if -9.9999999999999997e199 < (/.f64 z2 (*.f64 z3 z3)) < 2.0000000000000002e-5

   1. Initial program 81.3%

      \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}\right)\right)} \]

      6. add-to-fraction N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{z1 \cdot z1}}\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{\mathsf{neg}\left(z1 \cdot z1\right)}}} \]

      8. div-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.5%

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in z2 around 0

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.8%

      \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-z1\right) \cdot z1}{z0}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-z1\right) \cdot z1}}{z0} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(-z1\right) \cdot \frac{z1}{z0}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

      6. lower-/.f64 55.0%

         \[\leadsto \color{blue}{\frac{z1}{z0}} \cdot \left(-z1\right) \]

   8. Applied rewrites 55.0%

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

Alternative 9: 55.0% accurate, 2.5× speedup

Math

\[\frac{z1}{z0} \cdot \left(-z1\right) \]

FPCore

(FPCore (z2 z3 z0 z1)
  :precision binary64
  (* (/ z1 z0) (- z1)))

C

double code(double z2, double z3, double z0, double z1) {
	return (z1 / 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(z2, z3, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = (z1 / z0) * -z1
end function

Java

public static double code(double z2, double z3, double z0, double z1) {
	return (z1 / z0) * -z1;
}

Python

def code(z2, z3, z0, z1):
	return (z1 / z0) * -z1

Julia

function code(z2, z3, z0, z1)
	return Float64(Float64(z1 / z0) * Float64(-z1))
end

MATLAB

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

Wolfram

code[z2_, z3_, z0_, z1_] := N[(N[(z1 / z0), $MachinePrecision] * (-z1)), $MachinePrecision]

Derivation

1. Initial program 81.3%

   \[\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\frac{z2}{z3 \cdot z3} + \frac{z0}{z1 \cdot z1}\right)}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\left(\frac{z2}{z3 \cdot z3} + \color{blue}{\frac{z0}{z1 \cdot z1}}\right)\right)} \]

   6. add-to-fraction N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{z1 \cdot z1}}\right)} \]

   7. distribute-neg-frac2 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z2}{z3 \cdot z3} \cdot \left(z1 \cdot z1\right) + z0}{\mathsf{neg}\left(z1 \cdot z1\right)}}} \]

   8. div-flip-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 68.5%

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

3. Applied rewrites 68.5%

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

4. Taylor expanded in z2 around 0

   \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
5. Step-by-step derivation

6. Applied rewrites 50.8%

   \[\leadsto \frac{\left(-z1\right) \cdot z1}{\color{blue}{z0}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-z1\right) \cdot z1}{z0}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-z1\right) \cdot z1}}{z0} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(-z1\right) \cdot \frac{z1}{z0}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]

   6. lower-/.f64 55.0%

      \[\leadsto \color{blue}{\frac{z1}{z0}} \cdot \left(-z1\right) \]

8. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{z1}{z0} \cdot \left(-z1\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z2 z3 z0 z1)
  :name "(/ -1 (+ (/ z2 (* z3 z3)) (/ z0 (* z1 z1))))"
  :precision binary64
  (/ -1.0 (+ (/ z2 (* z3 z3)) (/ z0 (* z1 z1)))))