fabs fraction 1

Percentage Accurate: 91.6% → 99.6%

Time: 6.2s

Alternatives: 15

Speedup: 1.6×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-67}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1e-67)
   (fabs (/ (fma (- 1.0 z) x 4.0) 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 <= 1e-67) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / 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 <= 1e-67)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / 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, 1e-67], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $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 < 9.99999999999999943e-68

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.9%

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

   if 9.99999999999999943e-68 < y

   1. Initial program 96.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-67}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y}, \frac{4 + x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\left(-z\right) \cdot \frac{x}{y\_m}\right|\\ t_1 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-306}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (- z) (/ x y_m))))
        (t_1 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 1e-306)
       (fabs (/ (- x -4.0) y_m))
       (if (<= t_1 1e+303) (/ (fma (- 1.0 z) x 4.0) y_m) t_0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((-z * (x / y_m)));
	double t_1 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 1e-306) {
		tmp = fabs(((x - -4.0) / y_m));
	} else if (t_1 <= 1e+303) {
		tmp = fma((1.0 - z), x, 4.0) / y_m;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(-z) * Float64(x / y_m)))
	t_1 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 1e-306)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	elseif (t_1 <= 1e+303)
		tmp = Float64(fma(Float64(1.0 - z), x, 4.0) / y_m);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[((-z) * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 1e-306], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+303], N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -inf.0 or 1e303 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 75.4%

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1.00000000000000003e-306

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.1%

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

   if 1.00000000000000003e-306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1e303

   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

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

   5. Applied rewrites 97.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 95.7

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

   7. Applied rewrites 95.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -\infty:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{-306}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;t\_0 \leq 10^{-306}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

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 (- INFINITY))
     (fabs (* (- x) (/ z y_m)))
     (if (<= t_0 1e-306)
       (fabs (/ (- x -4.0) y_m))
       (if (<= t_0 INFINITY)
         (/ (fma (- 1.0 z) x 4.0) y_m)
         (fabs (/ 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 <= -((double) INFINITY)) {
		tmp = fabs((-x * (z / y_m)));
	} else if (t_0 <= 1e-306) {
		tmp = fabs(((x - -4.0) / y_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((1.0 - z), x, 4.0) / y_m;
	} else {
		tmp = fabs((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 <= Float64(-Inf))
		tmp = abs(Float64(Float64(-x) * Float64(z / y_m)));
	elseif (t_0 <= 1e-306)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(1.0 - z), x, 4.0) / y_m);
	else
		tmp = abs(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, (-Infinity)], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e-306], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.2%

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

   if -inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1.00000000000000003e-306

   1. Initial program 95.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.1%

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

   if 1.00000000000000003e-306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.3%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 96.7

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

   7. Applied rewrites 96.7%

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

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -\infty:\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{-306}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-227}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;t\_0 \leq 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

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 -1e-227)
     (fabs (* (- 1.0 z) (/ x y_m)))
     (if (<= t_0 1e+303)
       (/ (fma (- 1.0 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 <= -1e-227) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else if (t_0 <= 1e+303) {
		tmp = fma((1.0 - 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 <= -1e-227)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	elseif (t_0 <= 1e+303)
		tmp = Float64(fma(Float64(1.0 - z), 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, -1e-227], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+303], N[(N[(N[(1.0 - z), $MachinePrecision] * 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)) < -9.99999999999999945e-228

   1. Initial program 99.0%

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

   5. Applied rewrites 59.7%

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

   if -9.99999999999999945e-228 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1e303

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.0%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 88.8

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

   7. Applied rewrites 88.8%

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

   if 1e303 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 64.4%

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

   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. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -1 \cdot 10^{-227}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 10^{-306}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

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 1e-306)
     (fabs (/ (- x -4.0) y_m))
     (if (<= t_0 INFINITY) (/ (fma (- 1.0 z) x 4.0) y_m) (fabs (/ 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 <= 1e-306) {
		tmp = fabs(((x - -4.0) / y_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((1.0 - z), x, 4.0) / y_m;
	} else {
		tmp = fabs((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 <= 1e-306)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(1.0 - z), x, 4.0) / y_m);
	else
		tmp = abs(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, 1e-306], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(x / 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)) < 1.00000000000000003e-306

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 69.8%

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

   if 1.00000000000000003e-306 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.3%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 96.7

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

   7. Applied rewrites 96.7%

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

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{-306}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -5500000000 \lor \neg \left(z \leq 6 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -5500000000.0) (not (<= z 6e+23)))
   (/ (fma (- z) x 4.0) y_m)
   (fabs (/ (- x -4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -5500000000.0) || !(z <= 6e+23)) {
		tmp = fma(-z, x, 4.0) / y_m;
	} else {
		tmp = fabs(((x - -4.0) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -5500000000.0) || !(z <= 6e+23))
		tmp = Float64(fma(Float64(-z), x, 4.0) / y_m);
	else
		tmp = abs(Float64(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[z, -5500000000.0], N[Not[LessEqual[z, 6e+23]], $MachinePrecision]], N[(N[((-z) * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e9 or 6.0000000000000002e23 < z

   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

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

   5. Applied rewrites 96.0%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 53.3

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

   7. Applied rewrites 53.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 53.3%

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

   if -5.5e9 < z < 6.0000000000000002e23

   1. Initial program 94.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5500000000 \lor \neg \left(z \leq 6 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+77} \lor \neg \left(z \leq 2.12 \cdot 10^{+127}\right):\\ \;\;\;\;\frac{-z}{y\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.32e+77) (not (<= z 2.12e+127)))
   (* (/ (- z) y_m) x)
   (fabs (/ (- x -4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.32e+77) || !(z <= 2.12e+127)) {
		tmp = (-z / y_m) * x;
	} else {
		tmp = fabs(((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 ((z <= (-1.32d+77)) .or. (.not. (z <= 2.12d+127))) then
        tmp = (-z / y_m) * x
    else
        tmp = abs(((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 ((z <= -1.32e+77) || !(z <= 2.12e+127)) {
		tmp = (-z / y_m) * x;
	} else {
		tmp = Math.abs(((x - -4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.32e+77) or not (z <= 2.12e+127):
		tmp = (-z / y_m) * x
	else:
		tmp = math.fabs(((x - -4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.32e+77) || !(z <= 2.12e+127))
		tmp = Float64(Float64(Float64(-z) / y_m) * x);
	else
		tmp = abs(Float64(Float64(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 ((z <= -1.32e+77) || ~((z <= 2.12e+127)))
		tmp = (-z / y_m) * x;
	else
		tmp = abs(((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[z, -1.32e+77], N[Not[LessEqual[z, 2.12e+127]], $MachinePrecision]], N[(N[((-z) / y$95$m), $MachinePrecision] * x), $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e77 or 2.12000000000000011e127 < z

   1. Initial program 86.7%

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

   5. Applied rewrites 78.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 49.7

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

   7. Applied rewrites 49.7%

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

   9. Applied rewrites 46.5%

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

   if -1.32e77 < z < 2.12000000000000011e127

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+77} \lor \neg \left(z \leq 2.12 \cdot 10^{+127}\right):\\ \;\;\;\;\frac{-z}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y\_m}\\ \mathbf{elif}\;z \leq 2.12 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{y\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.32e+77)
   (/ (* (- z) x) y_m)
   (if (<= z 2.12e+127) (fabs (/ (- x -4.0) y_m)) (* (/ (- z) y_m) x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.32e+77) {
		tmp = (-z * x) / y_m;
	} else if (z <= 2.12e+127) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = (-z / y_m) * x;
	}
	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 <= (-1.32d+77)) then
        tmp = (-z * x) / y_m
    else if (z <= 2.12d+127) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = (-z / y_m) * x
    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 <= -1.32e+77) {
		tmp = (-z * x) / y_m;
	} else if (z <= 2.12e+127) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = (-z / y_m) * x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.32e+77:
		tmp = (-z * x) / y_m
	elif z <= 2.12e+127:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = (-z / y_m) * x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.32e+77)
		tmp = Float64(Float64(Float64(-z) * x) / y_m);
	elseif (z <= 2.12e+127)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = Float64(Float64(Float64(-z) / y_m) * x);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.32e+77)
		tmp = (-z * x) / y_m;
	elseif (z <= 2.12e+127)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = (-z / y_m) * x;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.32e+77], N[(N[((-z) * x), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[z, 2.12e+127], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[(N[((-z) / y$95$m), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e77

   1. Initial program 92.2%

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

   5. Applied rewrites 79.4%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 50.9

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

   7. Applied rewrites 50.9%

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

   9. Applied rewrites 48.9%

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

   if -1.32e77 < z < 2.12000000000000011e127

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.4%

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

   if 2.12000000000000011e127 < z

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 77.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 48.0

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

   7. Applied rewrites 48.0%

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

   9. Applied rewrites 42.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y}\\ \mathbf{elif}\;z \leq 2.12 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3150000:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - z}{y\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 3150000.0) (fabs (/ (- x -4.0) y_m)) (* (/ (- 1.0 z) y_m) x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 3150000.0) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = ((1.0 - z) / y_m) * x;
	}
	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 <= 3150000.0d0) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = ((1.0d0 - z) / y_m) * x
    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 <= 3150000.0) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = ((1.0 - z) / y_m) * x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 3150000.0:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = ((1.0 - z) / y_m) * x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 3150000.0)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = Float64(Float64(Float64(1.0 - z) / y_m) * x);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 3150000.0)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = ((1.0 - z) / y_m) * x;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 3150000.0], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[(N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.15e6

   1. Initial program 90.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 72.2%

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

   if 3.15e6 < x

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.4%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
   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 2.8

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 46.4%

       \[\leadsto \color{blue}{\frac{1 - z}{y} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3150000:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - z}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.55) (fabs (/ 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 <= -1.55) {
		tmp = fabs((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 <= (-1.55d0)) then
        tmp = abs((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 <= -1.55) {
		tmp = Math.abs((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 <= -1.55:
		tmp = math.fabs((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 <= -1.55)
		tmp = abs(Float64(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 <= -1.55)
		tmp = abs((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, -1.55], N[Abs[N[(x / y$95$m), $MachinePrecision]], $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 < -1.55000000000000004

   1. Initial program 84.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 69.5%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 93.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.3%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
   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 35.1

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

   7. Applied rewrites 35.1%

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

   if 4 < x

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 58.9%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 26.7

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

   13. Applied rewrites 26.7%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.2% accurate, 1.6× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (fma (- 1.0 z) x 4.0) y_m)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.1%

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

6. Final simplification 98.1%

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

Alternative 12: 69.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = fabs((x / y_m));
	} else {
		tmp = (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 <= (-4.0d0)) then
        tmp = abs((x / y_m))
    else
        tmp = (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 <= -4.0) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = (x - -4.0) / y_m;
	}
	return tmp;
}

Python

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

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = abs(Float64(x / y_m));
	else
		tmp = Float64(Float64(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 <= -4.0)
		tmp = abs((x / y_m));
	else
		tmp = (x - -4.0) / 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[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 84.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 69.5%

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

   if -4 < x

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 49.8%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
   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 23.9

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

   7. Applied rewrites 23.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 32.2%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.4% 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 90.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.4%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
   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 24.1

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

   7. Applied rewrites 24.1%

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

   if 4 < x

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 58.9%

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

       1. lift-fabs.f64 N/A

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

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

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 26.7

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

   13. Applied rewrites 26.7%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 69.0%

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

6. Final simplification 69.0%

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

Alternative 15: 18.2% 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 90.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 64.2%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 35.5%

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

    1. lift-fabs.f64 N/A

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

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

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

    3. sqrt-prod N/A

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

    4. rem-square-sqrt 19.0

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

13. Applied rewrites 19.0%

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

14. Final simplification 19.0%

    \[\leadsto \frac{x}{y} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))