fabs fraction 1

Percentage Accurate: 91.9% → 99.7%

Time: 7.5s

Alternatives: 13

Speedup: 1.0×

• 21.2% 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: 91.9% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6000 \lor \neg \left(x \leq 4 \cdot 10^{+28}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6000.0) (not (<= x 4e+28)))
   (fabs (* (- 1.0 z) (/ x y)))
   (fabs (/ (fma (- 1.0 z) x 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6000.0) || !(x <= 4e+28)) {
		tmp = fabs(((1.0 - z) * (x / y)));
	} else {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6000.0) || !(x <= 4e+28))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	else
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6000.0], N[Not[LessEqual[x, 4e+28]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6e3 or 3.99999999999999983e28 < x

   1. Initial program 85.5%

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

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

   if -6e3 < x < 3.99999999999999983e28

   1. Initial program 95.9%

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

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6000 \lor \neg \left(x \leq 4 \cdot 10^{+28}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \frac{x}{y} \cdot z\\ \mathbf{if}\;t\_0 - t\_1 \leq -1000:\\ \;\;\;\;\left|t\_1 - t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)) (t_1 (* (/ x y) z)))
   (if (<= (- t_0 t_1) -1000.0)
     (fabs (- t_1 t_0))
     (fabs (/ (fma (- 1.0 z) x 4.0) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = (x / y) * z;
	double tmp;
	if ((t_0 - t_1) <= -1000.0) {
		tmp = fabs((t_1 - t_0));
	} else {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	t_1 = Float64(Float64(x / y) * z)
	tmp = 0.0
	if (Float64(t_0 - t_1) <= -1000.0)
		tmp = abs(Float64(t_1 - t_0));
	else
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - t$95$1), $MachinePrecision], -1000.0], N[Abs[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1e3

   1. Initial program 99.9%

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

   if -1e3 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 87.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 97.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -1000:\\ \;\;\;\;\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y}, \frac{4 + x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e-50)
   (fabs (/ (fma (- 1.0 z) x 4.0) y))
   (fabs (fma (- x) (/ z y) (/ (+ 4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-50) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y));
	} else {
		tmp = fabs(fma(-x, (z / y), ((4.0 + x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e-50)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y));
	else
		tmp = abs(fma(Float64(-x), Float64(z / y), Float64(Float64(4.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e-50], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y), $MachinePrecision] + N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.99999999999999981e-50

   1. Initial program 87.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.8%

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

   if 5.99999999999999981e-50 < y

   1. Initial program 97.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 4: 83.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{\left(-x\right) \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 10^{-14}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.9e+113)
   (fabs (/ (* (- x) z) y))
   (if (<= z 1e-14) (fabs (/ (- x -4.0) y)) (fabs (* (- 1.0 z) (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.9e+113) {
		tmp = fabs(((-x * z) / y));
	} else if (z <= 1e-14) {
		tmp = fabs(((x - -4.0) / y));
	} else {
		tmp = fabs(((1.0 - z) * (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 <= (-5.9d+113)) then
        tmp = abs(((-x * z) / y))
    else if (z <= 1d-14) then
        tmp = abs(((x - (-4.0d0)) / y))
    else
        tmp = abs(((1.0d0 - z) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.9e+113) {
		tmp = Math.abs(((-x * z) / y));
	} else if (z <= 1e-14) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.9e+113:
		tmp = math.fabs(((-x * z) / y))
	elif z <= 1e-14:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.9e+113)
		tmp = abs(Float64(Float64(Float64(-x) * z) / y));
	elseif (z <= 1e-14)
		tmp = abs(Float64(Float64(x - -4.0) / y));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.9e+113)
		tmp = abs(((-x * z) / y));
	elseif (z <= 1e-14)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs(((1.0 - z) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.9e+113], N[Abs[N[(N[((-x) * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1e-14], N[Abs[N[(N[(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 3 regimes

2. if z < -5.90000000000000023e113

   1. Initial program 93.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 94.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 92.1%

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

   if -5.90000000000000023e113 < z < 9.99999999999999999e-15

   1. Initial program 93.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.7%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. *-inverses N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 94.6

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

   8. Applied rewrites 94.6%

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

   if 9.99999999999999999e-15 < z

   1. Initial program 80.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 5: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+113} \lor \neg \left(z \leq 3100000\right):\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.9e+113) (not (<= z 3100000.0)))
   (fabs (* (- z) (/ x y)))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.9e+113) || !(z <= 3100000.0)) {
		tmp = fabs((-z * (x / y)));
	} else {
		tmp = fabs(((x - -4.0) / 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 <= (-5.9d+113)) .or. (.not. (z <= 3100000.0d0))) then
        tmp = abs((-z * (x / y)))
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.9e+113) || !(z <= 3100000.0)) {
		tmp = Math.abs((-z * (x / y)));
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.9e+113) or not (z <= 3100000.0):
		tmp = math.fabs((-z * (x / y)))
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.9e+113) || !(z <= 3100000.0))
		tmp = abs(Float64(Float64(-z) * Float64(x / y)));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.9e+113) || ~((z <= 3100000.0)))
		tmp = abs((-z * (x / y)));
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.9e+113], N[Not[LessEqual[z, 3100000.0]], $MachinePrecision]], N[Abs[N[((-z) * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.90000000000000023e113 or 3.1e6 < z

   1. Initial program 87.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 84.6

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

   5. Applied rewrites 84.6%

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

   if -5.90000000000000023e113 < z < 3.1e6

   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 98.7%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. *-inverses N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 94.2

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

   8. Applied rewrites 94.2%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+113} \lor \neg \left(z \leq 3100000\right):\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{\left(-x\right) \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 3100000:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.9e+113)
   (fabs (/ (* (- x) z) y))
   (if (<= z 3100000.0) (fabs (/ (- x -4.0) y)) (fabs (* (- z) (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.9e+113) {
		tmp = fabs(((-x * z) / y));
	} else if (z <= 3100000.0) {
		tmp = fabs(((x - -4.0) / y));
	} else {
		tmp = fabs((-z * (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 <= (-5.9d+113)) then
        tmp = abs(((-x * z) / y))
    else if (z <= 3100000.0d0) then
        tmp = abs(((x - (-4.0d0)) / y))
    else
        tmp = abs((-z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.9e+113) {
		tmp = Math.abs(((-x * z) / y));
	} else if (z <= 3100000.0) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs((-z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.9e+113:
		tmp = math.fabs(((-x * z) / y))
	elif z <= 3100000.0:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs((-z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.9e+113)
		tmp = abs(Float64(Float64(Float64(-x) * z) / y));
	elseif (z <= 3100000.0)
		tmp = abs(Float64(Float64(x - -4.0) / y));
	else
		tmp = abs(Float64(Float64(-z) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.9e+113)
		tmp = abs(((-x * z) / y));
	elseif (z <= 3100000.0)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs((-z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.9e+113], N[Abs[N[(N[((-x) * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3100000.0], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-z) * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.90000000000000023e113

   1. Initial program 93.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 94.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 92.1%

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

   if -5.90000000000000023e113 < z < 3.1e6

   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 98.7%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. *-inverses N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if 3.1e6 < z

   1. Initial program 81.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

Alternative 7: 71.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+132} \lor \neg \left(z \leq 2.65 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e+132) (not (<= z 2.65e+73)))
   (* (/ (- x) y) z)
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+132) || !(z <= 2.65e+73)) {
		tmp = (-x / y) * z;
	} else {
		tmp = fabs(((x - -4.0) / 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 <= (-2.3d+132)) .or. (.not. (z <= 2.65d+73))) then
        tmp = (-x / y) * z
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+132) || !(z <= 2.65e+73)) {
		tmp = (-x / y) * z;
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.3e+132) or not (z <= 2.65e+73):
		tmp = (-x / y) * z
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e+132) || !(z <= 2.65e+73))
		tmp = Float64(Float64(Float64(-x) / y) * z);
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.3e+132) || ~((z <= 2.65e+73)))
		tmp = (-x / y) * z;
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e+132], N[Not[LessEqual[z, 2.65e+73]], $MachinePrecision]], N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000002e132 or 2.64999999999999998e73 < z

   1. Initial program 88.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 90.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 83.5

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

   8. Applied rewrites 83.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 52.0

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

   10. Applied rewrites 52.6%

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

   if -2.3000000000000002e132 < z < 2.64999999999999998e73

   1. Initial program 91.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.8%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. *-inverses N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 88.3

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

   8. Applied rewrites 88.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+132} \lor \neg \left(z \leq 2.65 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+132} \lor \neg \left(z \leq 3.9 \cdot 10^{+173}\right):\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e+132) (not (<= z 3.9e+173)))
   (* x (/ (- z) y))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+132) || !(z <= 3.9e+173)) {
		tmp = x * (-z / y);
	} else {
		tmp = fabs(((x - -4.0) / 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 <= (-2.3d+132)) .or. (.not. (z <= 3.9d+173))) then
        tmp = x * (-z / y)
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+132) || !(z <= 3.9e+173)) {
		tmp = x * (-z / y);
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.3e+132) or not (z <= 3.9e+173):
		tmp = x * (-z / y)
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e+132) || !(z <= 3.9e+173))
		tmp = Float64(x * Float64(Float64(-z) / y));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.3e+132) || ~((z <= 3.9e+173)))
		tmp = x * (-z / y);
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e+132], N[Not[LessEqual[z, 3.9e+173]], $MachinePrecision]], N[(x * N[((-z) / y), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000002e132 or 3.8999999999999998e173 < z

   1. Initial program 88.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 88.5

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

   5. Applied rewrites 88.5%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 53.7

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

   7. Applied rewrites 53.7%

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

   9. Applied rewrites 54.3%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 54.3%

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

   if -2.3000000000000002e132 < z < 3.8999999999999998e173

   1. Initial program 90.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 97.4%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. *-inverses N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 83.2

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

   8. Applied rewrites 83.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+132} \lor \neg \left(z \leq 3.9 \cdot 10^{+173}\right):\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5) (fabs (/ x y)) (if (<= x 4.0) (fabs (/ 4.0 y)) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = fabs((x / y));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = 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 (x <= (-1.5d0)) then
        tmp = abs((x / y))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = Math.abs((x / y));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5:
		tmp = math.fabs((x / y))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5)
		tmp = abs(Float64(x / y));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = abs((x / y));
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 68.5%

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

   if -1.5 < x < 4

   1. Initial program 95.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   if 4 < x

   1. Initial program 91.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 60.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 37.4

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

   10. Applied rewrites 37.4%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = fabs((x / y));
	} else {
		tmp = (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 (x <= (-4.0d0)) then
        tmp = abs((x / y))
    else
        tmp = (4.0d0 + x) / y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 68.5%

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

   if -4 < x

   1. Initial program 94.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.7%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. *-inverses N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower--.f64 65.1

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

   8. Applied rewrites 65.1%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 35.2

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

   10. Applied rewrites 35.2%

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

Alternative 11: 69.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fabs(((x - -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(((x - (-4.0d0)) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 95.8%

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

6. Taylor expanded in z around 0

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. div-add N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. lft-mult-inverse N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. fp-cancel-sign-sub N/A

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

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

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

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

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

   15. lft-mult-inverse N/A

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

   16. *-inverses N/A

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

   17. *-inverses N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower--.f64 66.4

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

8. Applied rewrites 66.4%

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

Alternative 12: 34.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fabs((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((x / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 71.3

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

5. Applied rewrites 71.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 40.0%

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

Alternative 13: 18.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = x / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 71.3

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

5. Applied rewrites 71.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 40.0%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 22.6

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

10. Applied rewrites 22.6%

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

Reproduce

herbie shell --seed 2024356 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))