fabs fraction 1

Percentage Accurate: 91.9% → 99.8%

Time: 3.4s

Alternatives: 12

Speedup: 1.0×

• 79.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{\left|y\right|}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{z}{\left|y\right|}, x, \frac{-4 - x}{\left|y\right|}\right)\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs y) 2e-48)
  (fabs (/ (fma z x (- -4.0 x)) (fabs y)))
  (fabs (fma (/ z (fabs y)) x (/ (- -4.0 x) (fabs y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 2e-48) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / fabs(y)));
	} else {
		tmp = fabs(fma((z / fabs(y)), x, ((-4.0 - x) / fabs(y))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 2e-48)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / abs(y)));
	else
		tmp = abs(fma(Float64(z / abs(y)), x, Float64(Float64(-4.0 - x) / abs(y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 2e-48], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z / N[Abs[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[(-4.0 - x), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-48

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   if 1.9999999999999999e-48 < y

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. metadata-eval N/A

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

      13. add-flip N/A

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

      14. distribute-frac-neg N/A

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

      15. lift-/.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l/ N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-/.f64 N/A

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

      24. distribute-frac-neg N/A

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

   5. Applied rewrites 94.4%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{if}\;x \leq -460:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7500000:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (* (/ (- 1.0 z) y) x))))
  (if (<= x -460.0)
    t_0
    (if (<= x 7500000.0) (fabs (/ (fma z x -4.0) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((1.0 - z) / y) * x));
	double tmp;
	if (x <= -460.0) {
		tmp = t_0;
	} else if (x <= 7500000.0) {
		tmp = fabs((fma(z, x, -4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(1.0 - z) / y) * x))
	tmp = 0.0
	if (x <= -460.0)
		tmp = t_0;
	elseif (x <= 7500000.0)
		tmp = abs(Float64(fma(z, x, -4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -460.0], t$95$0, If[LessEqual[x, 7500000.0], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -460 or 7.5e6 < x

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.3%

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

   7. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.3%

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 61.3%

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

   9. Applied rewrites 61.3%

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

   if -460 < x < 7.5e6

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 73.9%

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

Alternative 3: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{1} \cdot \frac{x}{y}\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 2e+14)
  (fabs (/ (fma z x (- -4.0 x)) y))
  (fabs (* (/ (- 1.0 z) 1.0) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+14) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y));
	} else {
		tmp = fabs((((1.0 - z) / 1.0) * (x / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+14)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y));
	else
		tmp = abs(Float64(Float64(Float64(1.0 - z) / 1.0) * Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+14], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / 1.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e14

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   if 2e14 < x

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.3%

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

   7. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 61.2%

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

   9. Applied rewrites 61.2%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= x 2.5e+27)
  (fabs (/ (fma z x (- -4.0 x)) y))
  (fabs (* (/ (- 1.0 z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.5e+27) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y));
	} else {
		tmp = fabs((((1.0 - z) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.5e+27)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y));
	else
		tmp = abs(Float64(Float64(Float64(1.0 - z) / y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.5e+27], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4999999999999999e27

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   if 2.4999999999999999e27 < x

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.3%

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

   7. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.3%

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 61.3%

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

   9. Applied rewrites 61.3%

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

Alternative 5: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{if}\;z \leq -440:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0047:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ (fma z x -4.0) y))))
  (if (<= z -440.0)
    t_0
    (if (<= z 0.0047) (fabs (/ (- -4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((fma(z, x, -4.0) / y));
	double tmp;
	if (z <= -440.0) {
		tmp = t_0;
	} else if (z <= 0.0047) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(fma(z, x, -4.0) / y))
	tmp = 0.0
	if (z <= -440.0)
		tmp = t_0;
	elseif (z <= 0.0047)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -440.0], t$95$0, If[LessEqual[z, 0.0047], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -440 or 0.0047000000000000002 < z

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 73.9%

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

   if -440 < z < 0.0047000000000000002

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. sub-flip N/A

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

      7. lift--.f64 70.6%

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

   8. Applied rewrites 70.6%

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

Alternative 6: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -460:\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-z}{\frac{y}{x}}\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= z -460.0)
  (fabs (* (- x) (/ z y)))
  (if (<= z 3.9e+37)
    (fabs (/ (- -4.0 x) y))
    (fabs (/ (- z) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -460.0) {
		tmp = fabs((-x * (z / y)));
	} else if (z <= 3.9e+37) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((-z / (y / x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-460.0d0)) then
        tmp = abs((-x * (z / y)))
    else if (z <= 3.9d+37) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((-z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -460.0) {
		tmp = Math.abs((-x * (z / y)));
	} else if (z <= 3.9e+37) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((-z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -460.0:
		tmp = math.fabs((-x * (z / y)))
	elif z <= 3.9e+37:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((-z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -460.0)
		tmp = abs(Float64(Float64(-x) * Float64(z / y)));
	elseif (z <= 3.9e+37)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(-z) / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -460.0)
		tmp = abs((-x * (z / y)));
	elseif (z <= 3.9e+37)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((-z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -460.0], N[Abs[N[((-x) * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.9e+37], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-z) / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -460

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-/.f64 39.4%

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

   6. Applied rewrites 39.4%

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

   if -460 < z < 3.8999999999999999e37

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. sub-flip N/A

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

      7. lift--.f64 70.6%

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

   8. Applied rewrites 70.6%

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

   if 3.8999999999999999e37 < z

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. distribute-neg-frac N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 42.1%

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

   6. Applied rewrites 42.1%

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

Alternative 7: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -460:\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-x}{y} \cdot z\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= z -460.0)
  (fabs (* (- x) (/ z y)))
  (if (<= z 3.9e+37)
    (fabs (/ (- -4.0 x) y))
    (fabs (* (/ (- x) y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -460.0) {
		tmp = fabs((-x * (z / y)));
	} else if (z <= 3.9e+37) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((-x / y) * z));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-460.0d0)) then
        tmp = abs((-x * (z / y)))
    else if (z <= 3.9d+37) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((-x / y) * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -460.0) {
		tmp = Math.abs((-x * (z / y)));
	} else if (z <= 3.9e+37) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((-x / y) * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -460.0:
		tmp = math.fabs((-x * (z / y)))
	elif z <= 3.9e+37:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((-x / y) * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -460.0)
		tmp = abs(Float64(Float64(-x) * Float64(z / y)));
	elseif (z <= 3.9e+37)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(Float64(-x) / y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -460.0)
		tmp = abs((-x * (z / y)));
	elseif (z <= 3.9e+37)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((-x / y) * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -460.0], N[Abs[N[((-x) * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.9e+37], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -460

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-/.f64 39.4%

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

   6. Applied rewrites 39.4%

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

   if -460 < z < 3.8999999999999999e37

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. sub-flip N/A

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

      7. lift--.f64 70.6%

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

   8. Applied rewrites 70.6%

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

   if 3.8999999999999999e37 < z

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 37.3%

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

   6. Applied rewrites 37.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 42.9%

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

   8. Applied rewrites 42.9%

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

Alternative 8: 84.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -460:\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= z -460.0)
  (fabs (* (- x) (/ z y)))
  (if (<= z 3.9e+37) (fabs (/ (- -4.0 x) y)) (fabs (/ (* x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -460.0) {
		tmp = fabs((-x * (z / y)));
	} else if (z <= 3.9e+37) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((x * z) / y));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-460.0d0)) then
        tmp = abs((-x * (z / y)))
    else if (z <= 3.9d+37) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((x * z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -460.0) {
		tmp = Math.abs((-x * (z / y)));
	} else if (z <= 3.9e+37) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((x * z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -460.0:
		tmp = math.fabs((-x * (z / y)))
	elif z <= 3.9e+37:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((x * z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -460.0)
		tmp = abs(Float64(Float64(-x) * Float64(z / y)));
	elseif (z <= 3.9e+37)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(x * z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -460.0)
		tmp = abs((-x * (z / y)));
	elseif (z <= 3.9e+37)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((x * z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -460.0], N[Abs[N[((-x) * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.9e+37], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -460

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-/.f64 39.4%

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

   6. Applied rewrites 39.4%

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

   if -460 < z < 3.8999999999999999e37

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. sub-flip N/A

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

      7. lift--.f64 70.6%

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

   8. Applied rewrites 70.6%

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

   if 3.8999999999999999e37 < z

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

Alternative 9: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x \cdot z}{y}\right|\\ \mathbf{if}\;z \leq -460:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ (* x z) y))))
  (if (<= z -460.0)
    t_0
    (if (<= z 3.9e+37) (fabs (/ (- -4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x * z) / y));
	double tmp;
	if (z <= -460.0) {
		tmp = t_0;
	} else if (z <= 3.9e+37) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * z) / y))
    if (z <= (-460.0d0)) then
        tmp = t_0
    else if (z <= 3.9d+37) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((x * z) / y));
	double tmp;
	if (z <= -460.0) {
		tmp = t_0;
	} else if (z <= 3.9e+37) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x * z) / y))
	tmp = 0
	if z <= -460.0:
		tmp = t_0
	elif z <= 3.9e+37:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x * z) / y))
	tmp = 0.0
	if (z <= -460.0)
		tmp = t_0;
	elseif (z <= 3.9e+37)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x * z) / y));
	tmp = 0.0;
	if (z <= -460.0)
		tmp = t_0;
	elseif (z <= 3.9e+37)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -460.0], t$95$0, If[LessEqual[z, 3.9e+37], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -460 or 3.8999999999999999e37 < z

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

   if -460 < z < 3.8999999999999999e37

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.6%

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

   6. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. sub-flip N/A

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

      7. lift--.f64 70.6%

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

   8. Applied rewrites 70.6%

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

Alternative 10: 70.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. sub-div N/A

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

   10. lower-/.f64 N/A

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

   11. sub-flip N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

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

   16. sub-negate N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval 96.3%

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

3. Applied rewrites 96.3%

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

4. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 70.6%

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

6. Applied rewrites 70.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. mul-1-neg N/A

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

   6. sub-flip N/A

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

   7. lift--.f64 70.6%

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

8. Applied rewrites 70.6%

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

Alternative 11: 69.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -460:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7500000:\\ \;\;\;\;\left|\frac{-4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ x y))))
  (if (<= x -460.0) t_0 (if (<= x 7500000.0) (fabs (/ -4.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -460.0) {
		tmp = t_0;
	} else if (x <= 7500000.0) {
		tmp = fabs((-4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if (x <= (-460.0d0)) then
        tmp = t_0
    else if (x <= 7500000.0d0) then
        tmp = abs(((-4.0d0) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -460.0) {
		tmp = t_0;
	} else if (x <= 7500000.0) {
		tmp = Math.abs((-4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -460.0:
		tmp = t_0
	elif x <= 7500000.0:
		tmp = math.fabs((-4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -460.0)
		tmp = t_0;
	elseif (x <= 7500000.0)
		tmp = abs(Float64(-4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -460.0)
		tmp = t_0;
	elseif (x <= 7500000.0)
		tmp = abs((-4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -460.0], t$95$0, If[LessEqual[x, 7500000.0], N[Abs[N[(-4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -460 or 7.5e6 < x

   1. Initial program 91.9%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.3%

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

   4. Applied rewrites 37.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.3%

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 34.2%

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

   10. Applied rewrites 34.2%

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

   if -460 < x < 7.5e6

   1. Initial program 91.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. sub-flip N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 96.3%

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 40.7%

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

   6. Applied rewrites 40.7%

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

Alternative 12: 34.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 37.3%

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

4. Applied rewrites 37.3%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 61.3%

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

7. Applied rewrites 61.3%

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

8. Taylor expanded in z around 0

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

   1. lower-/.f64 34.2%

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

10. Applied rewrites 34.2%

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

Reproduce

herbie shell --seed 2025213 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))