fabs fraction 1

Percentage Accurate: 91.6% → 97.8%

Time: 6.3s

Alternatives: 13

Speedup: 1.2×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+55}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e+55)
   (fabs (/ (fma (- 1.0 z) x 4.0) y))
   (fabs (* (- 1.0 z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e+55) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	} else {
		tmp = fabs(((1.0 - z) * (x / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e+55)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e+55], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9e55

   1. Initial program 94.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 99.4%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   if 1.9e55 < x

   1. Initial program 81.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-out N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 N/A

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

      22. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 2: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+44} \lor \neg \left(x \leq 4.1\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.5e+44) (not (<= x 4.1)))
   (fabs (* (- 1.0 z) (/ x y)))
   (fabs (/ (fma (- z) x 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.5e+44) || !(x <= 4.1)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else {
		tmp = fabs((fma(-z, x, 4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.5e+44) || !(x <= 4.1))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	else
		tmp = abs(Float64(fma(Float64(-z), x, 4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.5e+44], N[Not[LessEqual[x, 4.1]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[((-z) * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999999e44 or 4.0999999999999996 < x

   1. Initial program 84.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-out N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 N/A

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

      22. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -3.4999999999999999e44 < x < 4.0999999999999996

   1. Initial program 98.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around inf

      \[\leadsto \left|\frac{\mathsf{fma}\left(-1 \cdot z, x, 4\right)}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+44} \lor \neg \left(x \leq 4.1\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-42} \lor \neg \left(x \leq 2.55 \cdot 10^{-87}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -z, 4\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.5e-42) (not (<= x 2.55e-87)))
   (fabs (* (- 1.0 z) (/ x y)))
   (/ (fma x (- z) 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e-42) || !(x <= 2.55e-87)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else {
		tmp = fma(x, -z, 4.0) / y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.5e-42) || !(x <= 2.55e-87))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	else
		tmp = Float64(fma(x, Float64(-z), 4.0) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.5e-42], N[Not[LessEqual[x, 2.55e-87]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * (-z) + 4.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e-42 or 2.55000000000000012e-87 < x

   1. Initial program 87.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-out N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 N/A

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

      22. lower-/.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -4.5e-42 < x < 2.55000000000000012e-87

   1. Initial program 98.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{4}}{y} - \frac{x}{y} \cdot z\right| \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

      \[\leadsto \left|\frac{\color{blue}{4}}{y} - \frac{x}{y} \cdot z\right| \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{4}{y} - \frac{x}{y} \cdot z}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{4}{y} - \color{blue}{\frac{x}{y} \cdot z}\right| \]

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

         \[\leadsto \left|\color{blue}{\frac{4}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z}\right| \]

      4. +-commutative N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z + \frac{4}{y}}\right| \]

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

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)} + \frac{4}{y}\right| \]

      6. *-commutative N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{z \cdot \frac{x}{y}}\right)\right) + \frac{4}{y}\right| \]

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

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{x}{y}} + \frac{4}{y}\right| \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\color{blue}{\left(-z\right)} \cdot \frac{x}{y} + \frac{4}{y}\right| \]

      9. lift-/.f64 N/A

         \[\leadsto \left|\left(-z\right) \cdot \color{blue}{\frac{x}{y}} + \frac{4}{y}\right| \]

      10. associate-*r/ N/A

          \[\leadsto \left|\color{blue}{\frac{\left(-z\right) \cdot x}{y}} + \frac{4}{y}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|\frac{\left(-z\right) \cdot x}{y} + \color{blue}{\frac{4}{y}}\right| \]

      12. div-add-rev N/A

          \[\leadsto \left|\color{blue}{\frac{\left(-z\right) \cdot x + 4}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(-z\right) \cdot x + 4}{y}}\right| \]

      14. lower-fma.f64 99.9

          \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(-z, x, 4\right)}}{y}\right| \]

   7. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right|} \]

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 54.9

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-z\right) \cdot x + 4}}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-z\right)} + 4}{y} \]

      7. lower-fma.f64 54.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -z, 4\right)}}{y} \]

   9. Applied rewrites 54.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-42} \lor \neg \left(x \leq 2.55 \cdot 10^{-87}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -z, 4\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (fma (/ x y) (- 1.0 z) (/ 4.0 y))))

C

double code(double x, double y, double z) {
	return fabs(fma((x / y), (1.0 - z), (4.0 / y)));
}

Julia

function code(x, y, z)
	return abs(fma(Float64(x / y), Float64(1.0 - z), Float64(4.0 / y)))
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   2. *-commutative N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

   3. *-commutative N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   4. distribute-lft-out-- N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

   5. associate-*r/ N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

   6. *-rgt-identity N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

   7. associate-/l* N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

   8. associate--l+ N/A

      \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

   9. associate-*r/ N/A

      \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

   11. div-add N/A

       \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

   12. div-sub N/A

       \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   13. lower-/.f64 N/A

       \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

5. Applied rewrites 97.7%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \left|\mathsf{fma}\left(\frac{x}{y}, \color{blue}{1 - z}, \frac{4}{y}\right)\right| \]
8. Add Preprocessing

Alternative 5: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -64000000000 \lor \neg \left(z \leq 7 \cdot 10^{+83}\right):\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -64000000000.0) (not (<= z 7e+83)))
   (fabs (* (- z) (/ x y)))
   (fabs (/ (+ 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -64000000000.0) || !(z <= 7e+83)) {
		tmp = fabs((-z * (x / y)));
	} else {
		tmp = fabs(((4.0 + x) / y));
	}
	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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-64000000000.0d0)) .or. (.not. (z <= 7d+83))) then
        tmp = abs((-z * (x / y)))
    else
        tmp = abs(((4.0d0 + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -64000000000.0) || !(z <= 7e+83)) {
		tmp = Math.abs((-z * (x / y)));
	} else {
		tmp = Math.abs(((4.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -64000000000.0) or not (z <= 7e+83):
		tmp = math.fabs((-z * (x / y)))
	else:
		tmp = math.fabs(((4.0 + x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -64000000000.0) || !(z <= 7e+83))
		tmp = abs(Float64(Float64(-z) * Float64(x / y)));
	else
		tmp = abs(Float64(Float64(4.0 + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -64000000000.0) || ~((z <= 7e+83)))
		tmp = abs((-z * (x / y)));
	else
		tmp = abs(((4.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -64000000000.0], N[Not[LessEqual[z, 7e+83]], $MachinePrecision]], N[Abs[N[((-z) * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4e10 or 6.99999999999999954e83 < z

   1. Initial program 91.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \frac{x}{y}\right| \]

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if -6.4e10 < z < 6.99999999999999954e83

   1. Initial program 93.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around 0

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 93.6%

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -64000000000 \lor \neg \left(z \leq 7 \cdot 10^{+83}\right):\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -64000000000 \lor \neg \left(z \leq 7 \cdot 10^{+83}\right):\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -64000000000.0) (not (<= z 7e+83)))
   (fabs (* (- x) (/ z y)))
   (fabs (/ (+ 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -64000000000.0) || !(z <= 7e+83)) {
		tmp = fabs((-x * (z / y)));
	} else {
		tmp = fabs(((4.0 + x) / y));
	}
	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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-64000000000.0d0)) .or. (.not. (z <= 7d+83))) then
        tmp = abs((-x * (z / y)))
    else
        tmp = abs(((4.0d0 + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -64000000000.0) || !(z <= 7e+83)) {
		tmp = Math.abs((-x * (z / y)));
	} else {
		tmp = Math.abs(((4.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -64000000000.0) or not (z <= 7e+83):
		tmp = math.fabs((-x * (z / y)))
	else:
		tmp = math.fabs(((4.0 + x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -64000000000.0) || !(z <= 7e+83))
		tmp = abs(Float64(Float64(-x) * Float64(z / y)));
	else
		tmp = abs(Float64(Float64(4.0 + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -64000000000.0) || ~((z <= 7e+83)))
		tmp = abs((-x * (z / y)));
	else
		tmp = abs(((4.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -64000000000.0], N[Not[LessEqual[z, 7e+83]], $MachinePrecision]], N[Abs[N[((-x) * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4e10 or 6.99999999999999954e83 < z

   1. Initial program 91.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \frac{x}{y}\right| \]

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.3%

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

   if -6.4e10 < z < 6.99999999999999954e83

   1. Initial program 93.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 99.9%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around 0

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 93.6%

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -64000000000 \lor \neg \left(z \leq 7 \cdot 10^{+83}\right):\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+161} \lor \neg \left(z \leq 5 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -z, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.6e+161) (not (<= z 5e+28)))
   (/ (fma x (- z) 4.0) y)
   (fabs (/ (+ 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.6e+161) || !(z <= 5e+28)) {
		tmp = fma(x, -z, 4.0) / y;
	} else {
		tmp = fabs(((4.0 + x) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.6e+161) || !(z <= 5e+28))
		tmp = Float64(fma(x, Float64(-z), 4.0) / y);
	else
		tmp = abs(Float64(Float64(4.0 + x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.6e+161], N[Not[LessEqual[z, 5e+28]], $MachinePrecision]], N[(N[(x * (-z) + 4.0), $MachinePrecision] / y), $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5999999999999998e161 or 4.99999999999999957e28 < z

   1. Initial program 90.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{4}}{y} - \frac{x}{y} \cdot z\right| \]
   4. Step-by-step derivation

   5. Applied rewrites 98.6%

      \[\leadsto \left|\frac{\color{blue}{4}}{y} - \frac{x}{y} \cdot z\right| \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{4}{y} - \frac{x}{y} \cdot z}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{4}{y} - \color{blue}{\frac{x}{y} \cdot z}\right| \]

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

         \[\leadsto \left|\color{blue}{\frac{4}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z}\right| \]

      4. +-commutative N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z + \frac{4}{y}}\right| \]

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

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)} + \frac{4}{y}\right| \]

      6. *-commutative N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(\color{blue}{z \cdot \frac{x}{y}}\right)\right) + \frac{4}{y}\right| \]

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

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{x}{y}} + \frac{4}{y}\right| \]

      8. lift-neg.f64 N/A

         \[\leadsto \left|\color{blue}{\left(-z\right)} \cdot \frac{x}{y} + \frac{4}{y}\right| \]

      9. lift-/.f64 N/A

         \[\leadsto \left|\left(-z\right) \cdot \color{blue}{\frac{x}{y}} + \frac{4}{y}\right| \]

      10. associate-*r/ N/A

          \[\leadsto \left|\color{blue}{\frac{\left(-z\right) \cdot x}{y}} + \frac{4}{y}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|\frac{\left(-z\right) \cdot x}{y} + \color{blue}{\frac{4}{y}}\right| \]

      12. div-add-rev N/A

          \[\leadsto \left|\color{blue}{\frac{\left(-z\right) \cdot x + 4}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(-z\right) \cdot x + 4}{y}}\right| \]

      14. lower-fma.f64 95.5

          \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(-z, x, 4\right)}}{y}\right| \]

   7. Applied rewrites 95.5%

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

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right|} \]

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 46.0

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-z\right) \cdot x + 4}}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-z\right)} + 4}{y} \]

      7. lower-fma.f64 46.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -z, 4\right)}}{y} \]

   9. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, -z, 4\right)}{y}} \]

   if -2.5999999999999998e161 < z < 4.99999999999999957e28

   1. Initial program 93.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 98.8%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around 0

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 86.7%

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+161} \lor \neg \left(z \leq 5 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -z, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{\left(1 - z\right) \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.8e+249) (/ (* (- 1.0 z) x) y) (fabs (/ (+ 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+249) {
		tmp = ((1.0 - z) * x) / y;
	} else {
		tmp = fabs(((4.0 + x) / y));
	}
	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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8.8d+249)) then
        tmp = ((1.0d0 - z) * x) / y
    else
        tmp = abs(((4.0d0 + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+249) {
		tmp = ((1.0 - z) * x) / y;
	} else {
		tmp = Math.abs(((4.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.8e+249:
		tmp = ((1.0 - z) * x) / y
	else:
		tmp = math.fabs(((4.0 + x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.8e+249)
		tmp = Float64(Float64(Float64(1.0 - z) * x) / y);
	else
		tmp = abs(Float64(Float64(4.0 + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.8e+249)
		tmp = ((1.0 - z) * x) / y;
	else
		tmp = abs(((4.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.8e+249], N[(N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.7999999999999993e249

   1. Initial program 99.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

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

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

      17. distribute-lft-neg-out N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 N/A

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

      22. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 43.1

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

   9. Applied rewrites 43.1%

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

   if -8.7999999999999993e249 < z

   1. Initial program 92.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 97.5%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around 0

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 73.1%

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.8e+249) (* (/ (- x) y) z) (fabs (/ (+ 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+249) {
		tmp = (-x / y) * z;
	} else {
		tmp = fabs(((4.0 + x) / y));
	}
	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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8.8d+249)) then
        tmp = (-x / y) * z
    else
        tmp = abs(((4.0d0 + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+249) {
		tmp = (-x / y) * z;
	} else {
		tmp = Math.abs(((4.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.8e+249:
		tmp = (-x / y) * z
	else:
		tmp = math.fabs(((4.0 + x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.8e+249)
		tmp = Float64(Float64(Float64(-x) / y) * z);
	else
		tmp = abs(Float64(Float64(4.0 + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.8e+249)
		tmp = (-x / y) * z;
	else
		tmp = abs(((4.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.8e+249], N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.7999999999999993e249

   1. Initial program 99.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \frac{x}{y}\right| \]

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 42.8

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

   7. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\frac{-x}{y} \cdot z} \]

   if -8.7999999999999993e249 < z

   1. Initial program 92.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 97.5%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around 0

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 73.1%

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{-z}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.8e+249) (* (/ (- z) y) x) (fabs (/ (+ 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+249) {
		tmp = (-z / y) * x;
	} else {
		tmp = fabs(((4.0 + x) / y));
	}
	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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8.8d+249)) then
        tmp = (-z / y) * x
    else
        tmp = abs(((4.0d0 + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+249) {
		tmp = (-z / y) * x;
	} else {
		tmp = Math.abs(((4.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.8e+249:
		tmp = (-z / y) * x
	else:
		tmp = math.fabs(((4.0 + x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.8e+249)
		tmp = Float64(Float64(Float64(-z) / y) * x);
	else
		tmp = abs(Float64(Float64(4.0 + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.8e+249)
		tmp = (-z / y) * x;
	else
		tmp = abs(((4.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.8e+249], N[(N[((-z) / y), $MachinePrecision] * x), $MachinePrecision], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.7999999999999993e249

   1. Initial program 99.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \frac{x}{y}\right| \]

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.4%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 23.5

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

   10. Applied rewrites 23.5%

       \[\leadsto \color{blue}{\frac{-z}{y} \cdot x} \]

   if -8.7999999999999993e249 < z

   1. Initial program 92.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

      3. *-commutative N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      4. distribute-lft-out-- N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

      5. associate-*r/ N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

      6. *-rgt-identity N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

      7. associate-/l* N/A

         \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

      8. associate--l+ N/A

         \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

      9. associate-*r/ N/A

         \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

      11. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

      12. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

      13. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   5. Applied rewrites 97.5%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

   6. Taylor expanded in z around 0

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 73.1%

      \[\leadsto \left|\frac{4 + x}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{4 + x}{y}\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (/ (+ 4.0 x) y)))

C

double code(double x, double y, double z) {
	return fabs(((4.0 + x) / y));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs(((4.0d0 + x) / y))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs(((4.0 + x) / y));
}

Python

def code(x, y, z):
	return math.fabs(((4.0 + x) / y))

Julia

function code(x, y, z)
	return abs(Float64(Float64(4.0 + x) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs(((4.0 + x) / y));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   2. *-commutative N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

   3. *-commutative N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   4. distribute-lft-out-- N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

   5. associate-*r/ N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

   6. *-rgt-identity N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

   7. associate-/l* N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

   8. associate--l+ N/A

      \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

   9. associate-*r/ N/A

      \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

   11. div-add N/A

       \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

   12. div-sub N/A

       \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   13. lower-/.f64 N/A

       \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

5. Applied rewrites 97.7%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

6. Taylor expanded in z around 0

   \[\leadsto \left|\frac{4 + x}{y}\right| \]
7. Step-by-step derivation

8. Applied rewrites 70.1%

   \[\leadsto \left|\frac{4 + x}{y}\right| \]
9. Add Preprocessing

Alternative 12: 35.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{4 + x}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ 4.0 x) y))

C

double code(double x, double y, double z) {
	return (4.0 + x) / y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 + x) / y
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 + x) / y;
}

Python

def code(x, y, z):
	return (4.0 + x) / y

Julia

function code(x, y, z)
	return Float64(Float64(4.0 + x) / y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 + x) / y;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right) + 4 \cdot \frac{1}{y}}\right| \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   2. *-commutative N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x}\right| \]

   3. *-commutative N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   4. distribute-lft-out-- N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \color{blue}{\left(x \cdot \frac{1}{y} - x \cdot \frac{z}{y}\right)}\right| \]

   5. associate-*r/ N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\color{blue}{\frac{x \cdot 1}{y}} - x \cdot \frac{z}{y}\right)\right| \]

   6. *-rgt-identity N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{\color{blue}{x}}{y} - x \cdot \frac{z}{y}\right)\right| \]

   7. associate-/l* N/A

      \[\leadsto \left|4 \cdot \frac{1}{y} + \left(\frac{x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right)\right| \]

   8. associate--l+ N/A

      \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}}\right| \]

   9. associate-*r/ N/A

      \[\leadsto \left|\left(\color{blue}{\frac{4 \cdot 1}{y}} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\left(\frac{\color{blue}{4}}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right| \]

   11. div-add N/A

       \[\leadsto \left|\color{blue}{\frac{4 + x}{y}} - \frac{x \cdot z}{y}\right| \]

   12. div-sub N/A

       \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

   13. lower-/.f64 N/A

       \[\leadsto \left|\color{blue}{\frac{\left(4 + x\right) - x \cdot z}{y}}\right| \]

5. Applied rewrites 97.7%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}}\right| \]

6. Taylor expanded in z around 0

   \[\leadsto \left|\frac{4 + x}{y}\right| \]
7. Step-by-step derivation

8. Applied rewrites 70.1%

   \[\leadsto \left|\frac{4 + x}{y}\right| \]
9. Step-by-step derivation

   1. lift-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{4 + x}{y}\right|} \]

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

      \[\leadsto \color{blue}{\sqrt{\frac{4 + x}{y} \cdot \frac{4 + x}{y}}} \]

   3. sqrt-prod N/A

      \[\leadsto \color{blue}{\sqrt{\frac{4 + x}{y}} \cdot \sqrt{\frac{4 + x}{y}}} \]

   4. rem-square-sqrt 37.9

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]

10. Applied rewrites 37.9%

    \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
11. Add Preprocessing

Alternative 13: 40.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{4}{y}\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

double code(double x, double y, double z) {
	return fabs((4.0 / y));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((4.0d0 / y))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((4.0 / y));
}

Python

def code(x, y, z):
	return math.fabs((4.0 / y))

Julia

function code(x, y, z)
	return abs(Float64(4.0 / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((4.0 / y));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
4. Step-by-step derivation

   1. lower-/.f64 43.6

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]

5. Applied rewrites 43.6%

   \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))