fabs fraction 1

Percentage Accurate: 92.0% → 99.2%

Time: 6.9s

Alternatives: 13

Speedup: 1.2×

• 20.1% 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: 92.0% 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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e+106)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (fma (- x) (/ z y_m) (/ (+ 4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e+106) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(fma(-x, (z / y_m), ((4.0 + x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e+106)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(fma(Float64(-x), Float64(z / y_m), Float64(Float64(4.0 + x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5e+106], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision] + N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e106

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.5%

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

   if 4.9999999999999998e106 < y

   1. Initial program 96.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 5e-306)
     (/ (fma z x (- -4.0 x)) y_m)
     (if (<= t_0 2e+301)
       (/ (fma z (- x) 4.0) y_m)
       (fabs (* (- z) (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = fma(z, x, (-4.0 - x)) / y_m;
	} else if (t_0 <= 2e+301) {
		tmp = fma(z, -x, 4.0) / y_m;
	} else {
		tmp = fabs((-z * (x / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= 5e-306)
		tmp = Float64(fma(z, x, Float64(-4.0 - x)) / y_m);
	elseif (t_0 <= 2e+301)
		tmp = Float64(fma(z, Float64(-x), 4.0) / y_m);
	else
		tmp = abs(Float64(Float64(-z) * Float64(x / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-306], N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e+301], N[(N[(z * (-x) + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[((-z) * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.99999999999999998e-306

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 94.3%

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

   if 4.99999999999999998e-306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 2.00000000000000011e301

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.5

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

   7. Applied rewrites 71.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

   9. Applied rewrites 71.4%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 70.0

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

       5. lift-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 70.0

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

   11. Applied rewrites 70.0%

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

   if 2.00000000000000011e301 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 46.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 5e-306)
     (/ (fma z x (- -4.0 x)) y_m)
     (if (<= t_0 INFINITY)
       (/ (fma z (- x) 4.0) y_m)
       (fabs (/ (- -4.0 x) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = fma(z, x, (-4.0 - x)) / y_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(z, -x, 4.0) / y_m;
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= 5e-306)
		tmp = Float64(fma(z, x, Float64(-4.0 - x)) / y_m);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(z, Float64(-x), 4.0) / y_m);
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-306], N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(z * (-x) + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.99999999999999998e-306

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 94.3%

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

   if 4.99999999999999998e-306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.1

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

   7. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

   9. Applied rewrites 71.0%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 69.7

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

       5. lift-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 69.7

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

   11. Applied rewrites 69.7%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 100.0%

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

Alternative 4: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, -x\right)}{y\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 5e-306)
     (/ (fma z x (- x)) y_m)
     (if (<= t_0 INFINITY)
       (/ (fma z (- x) 4.0) y_m)
       (fabs (/ (- -4.0 x) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = fma(z, x, -x) / y_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(z, -x, 4.0) / y_m;
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= 5e-306)
		tmp = Float64(fma(z, x, Float64(-x)) / y_m);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(z, Float64(-x), 4.0) / y_m);
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-306], N[(N[(z * x + (-x)), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(z * (-x) + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.99999999999999998e-306

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.8%

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

   10. Applied rewrites 48.4%

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

   if 4.99999999999999998e-306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 71.1

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

   7. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

   9. Applied rewrites 71.0%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 69.7

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

       5. lift-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 69.7

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

   11. Applied rewrites 69.7%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 100.0%

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

Alternative 5: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.52) (not (<= x 4.2)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (- (* z x) 4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.52) || !(x <= 4.2)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs((((z * x) - 4.0) / y_m));
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.52d0)) .or. (.not. (x <= 4.2d0))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs((((z * x) - 4.0d0) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.52) || !(x <= 4.2)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs((((z * x) - 4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.52) or not (x <= 4.2):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs((((z * x) - 4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.52) || !(x <= 4.2))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(Float64(z * x) - 4.0) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.52) || ~((x <= 4.2)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs((((z * x) - 4.0) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.52], N[Not[LessEqual[x, 4.2]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(z * x), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.52 or 4.20000000000000018 < x

   1. Initial program 86.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   if -1.52 < x < 4.20000000000000018

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-54} \lor \neg \left(x \leq 7.5 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -4.2e-54) (not (<= x 7.5e-6)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (/ (fma z (- x) 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -4.2e-54) || !(x <= 7.5e-6)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fma(z, -x, 4.0) / y_m;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -4.2e-54) || !(x <= 7.5e-6))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = Float64(fma(z, Float64(-x), 4.0) / y_m);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -4.2e-54], N[Not[LessEqual[x, 7.5e-6]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(z * (-x) + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e-54 or 7.50000000000000019e-6 < x

   1. Initial program 87.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   if -4.2e-54 < x < 7.50000000000000019e-6

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.7

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

   7. Applied rewrites 93.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

   9. Applied rewrites 99.4%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 48.7

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

       5. lift-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 48.7

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

   11. Applied rewrites 48.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-54} \lor \neg \left(x \leq 7.5 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-40} \lor \neg \left(x \leq 0.00025\right):\\ \;\;\;\;\left|\frac{z - 1}{y\_m} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -7e-40) (not (<= x 0.00025)))
   (fabs (* (/ (- z 1.0) y_m) x))
   (/ (fma z (- x) 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -7e-40) || !(x <= 0.00025)) {
		tmp = fabs((((z - 1.0) / y_m) * x));
	} else {
		tmp = fma(z, -x, 4.0) / y_m;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -7e-40) || !(x <= 0.00025))
		tmp = abs(Float64(Float64(Float64(z - 1.0) / y_m) * x));
	else
		tmp = Float64(fma(z, Float64(-x), 4.0) / y_m);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -7e-40], N[Not[LessEqual[x, 0.00025]], $MachinePrecision]], N[Abs[N[(N[(N[(z - 1.0), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[(N[(z * (-x) + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000003e-40 or 2.5000000000000001e-4 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 97.6%

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

   if -7.0000000000000003e-40 < x < 2.5000000000000001e-4

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.3

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

   7. Applied rewrites 93.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

   9. Applied rewrites 99.5%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 48.0

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

       5. lift-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 48.0

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

   11. Applied rewrites 48.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-40} \lor \neg \left(x \leq 0.00025\right):\\ \;\;\;\;\left|\frac{z - 1}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -x, 4\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+65} \lor \neg \left(z \leq 1.5 \cdot 10^{+75}\right):\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -6.2e+65) (not (<= z 1.5e+75)))
   (fabs (* (/ z y_m) x))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -6.2e+65) || !(z <= 1.5e+75)) {
		tmp = fabs(((z / y_m) * x));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.2d+65)) .or. (.not. (z <= 1.5d+75))) then
        tmp = abs(((z / y_m) * x))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -6.2e+65) || !(z <= 1.5e+75)) {
		tmp = Math.abs(((z / y_m) * x));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -6.2e+65) or not (z <= 1.5e+75):
		tmp = math.fabs(((z / y_m) * x))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -6.2e+65) || !(z <= 1.5e+75))
		tmp = abs(Float64(Float64(z / y_m) * x));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -6.2e+65) || ~((z <= 1.5e+75)))
		tmp = abs(((z / y_m) * x));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -6.2e+65], N[Not[LessEqual[z, 1.5e+75]], $MachinePrecision]], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.19999999999999981e65 or 1.5e75 < z

   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

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 74.8%

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

   if -6.19999999999999981e65 < z < 1.5e75

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+65} \lor \neg \left(z \leq 1.5 \cdot 10^{+75}\right):\\ \;\;\;\;\left|\frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+27}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1e+27)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (* (- 1.0 z) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1e+27) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 1e+27)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 1e+27], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e27

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.9%

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

   if 1e27 < x

   1. Initial program 85.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 10: 69.4% accurate, 1.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0) (/ (- -4.0 x) y_m) (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-4.0 - x) / y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.0d0)) then
        tmp = ((-4.0d0) - x) / y_m
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -4.0], N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.3%

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

   10. Applied rewrites 21.7%

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

   if -4 < x < 4

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 73.9

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

   5. Applied rewrites 73.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 34.4

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

   7. Applied rewrites 34.4%

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

   if 4 < x

   1. Initial program 87.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]
   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 60.7

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto \frac{x}{\color{blue}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 39.7%

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

Alternative 11: 54.2% accurate, 2.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 54.4

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

   5. Applied rewrites 54.4%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 25.4

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

   7. Applied rewrites 25.4%

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

   if 4 < x

   1. Initial program 87.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]
   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 60.7

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto \frac{x}{\color{blue}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 39.7%

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

Alternative 12: 70.2% accurate, 2.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (- -4.0 x) y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 68.0%

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

Alternative 13: 17.7% accurate, 3.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ x y_m))

C

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

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x / y_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x / y$95$m), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in x around 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 60.6

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

5. Applied rewrites 60.6%

   \[\leadsto \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]
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 34.0

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

7. Applied rewrites 34.0%

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

9. Applied rewrites 34.3%

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

10. Taylor expanded in z around 0

    \[\leadsto \frac{x}{\color{blue}{y}} \]
11. Step-by-step derivation

12. Applied rewrites 21.2%

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

Reproduce

herbie shell --seed 2024364 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))