fabs fraction 1

Percentage Accurate: 91.7% → 98.0%

Time: 10.5s

Alternatives: 9

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    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: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs(((4.0d0 / y) + ((x / y) * (1.0d0 - z))))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs(((4.0 / y) + ((x / y) * (1.0 - z))));
}

Python

def code(x, y, z):
	return math.fabs(((4.0 / y) + ((x / y) * (1.0 - z))))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = abs(((4.0 / y) + ((x / y) * (1.0 - z))));
end

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

   11. associate-*l/ N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

   16. div-sub N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

   18. associate--r- N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

   21. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

   22. sub-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

3. Simplified 98.8%

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

Alternative 2: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y} \cdot z\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -10.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{-60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y) z))) (t_1 (fabs (/ x y))))
   (if (<= x -3.9e+151)
     t_0
     (if (<= x -10.5)
       t_1
       (if (<= x 1e-60)
         (fabs (/ 4.0 y))
         (if (<= x 0.0225)
           (fabs (* x (/ z y)))
           (if (<= x 2.9e+160) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x / y) * z));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -3.9e+151) {
		tmp = t_0;
	} else if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 1e-60) {
		tmp = fabs((4.0 / y));
	} else if (x <= 0.0225) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 2.9e+160) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(((x / y) * z))
    t_1 = abs((x / y))
    if (x <= (-3.9d+151)) then
        tmp = t_0
    else if (x <= (-10.5d0)) then
        tmp = t_1
    else if (x <= 1d-60) then
        tmp = abs((4.0d0 / y))
    else if (x <= 0.0225d0) then
        tmp = abs((x * (z / y)))
    else if (x <= 2.9d+160) then
        tmp = t_1
    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) * z));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -3.9e+151) {
		tmp = t_0;
	} else if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 1e-60) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 0.0225) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 2.9e+160) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x / y) * z))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -3.9e+151:
		tmp = t_0
	elif x <= -10.5:
		tmp = t_1
	elif x <= 1e-60:
		tmp = math.fabs((4.0 / y))
	elif x <= 0.0225:
		tmp = math.fabs((x * (z / y)))
	elif x <= 2.9e+160:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x / y) * z))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -3.9e+151)
		tmp = t_0;
	elseif (x <= -10.5)
		tmp = t_1;
	elseif (x <= 1e-60)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 0.0225)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 2.9e+160)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x / y) * z));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -3.9e+151)
		tmp = t_0;
	elseif (x <= -10.5)
		tmp = t_1;
	elseif (x <= 1e-60)
		tmp = abs((4.0 / y));
	elseif (x <= 0.0225)
		tmp = abs((x * (z / y)));
	elseif (x <= 2.9e+160)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.9e+151], t$95$0, If[LessEqual[x, -10.5], t$95$1, If[LessEqual[x, 1e-60], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0225], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.9e+160], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.89999999999999976e151 or 2.8999999999999999e160 < x

   1. Initial program 87.5%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\frac{{x}^{3} + {4}^{3}}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left({x}^{3} + {4}^{3}\right) \cdot \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(x \cdot z\right)\right)}{y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)}{y}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)\right), y\right)\right) \]

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \frac{x \cdot z}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{x \cdot z}{y}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \frac{x \cdot z}{y}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{x \cdot z}{y}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{z \cdot x}{y}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(z \cdot \frac{x}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 78.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right)\right) \]

   7. Simplified 78.3%

      \[\leadsto \left|\color{blue}{0 - z \cdot \frac{x}{y}}\right| \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. fabs-neg N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z}{y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right) \]

      7. *-lowering-*.f64 51.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right) \]

   9. Applied egg-rr 51.2%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right) \]

       4. /-lowering-/.f64 78.3%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   11. Applied egg-rr 78.3%

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

   if -3.89999999999999976e151 < x < -10.5 or 0.022499999999999999 < x < 2.8999999999999999e160

   1. Initial program 97.2%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 71.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   7. Simplified 71.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 66.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, y\right)\right) \]

   10. Simplified 66.9%

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

   if -10.5 < x < 9.9999999999999997e-61

   1. Initial program 97.4%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 76.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   7. Simplified 76.2%

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

   if 9.9999999999999997e-61 < x < 0.022499999999999999

   1. Initial program 99.8%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\frac{{x}^{3} + {4}^{3}}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left({x}^{3} + {4}^{3}\right) \cdot \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(x \cdot z\right)\right)}{y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)}{y}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)\right), y\right)\right) \]

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \frac{x \cdot z}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{x \cdot z}{y}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \frac{x \cdot z}{y}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{x \cdot z}{y}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{z \cdot x}{y}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(z \cdot \frac{x}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 73.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right)\right) \]

   7. Simplified 73.2%

      \[\leadsto \left|\color{blue}{0 - z \cdot \frac{x}{y}}\right| \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. fabs-neg N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z}{y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right) \]

      7. *-lowering-*.f64 72.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right) \]

   9. Applied egg-rr 72.9%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{z}{y}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{y} \cdot x\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{z}{y}\right), x\right)\right) \]

       4. /-lowering-/.f64 73.3%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), x\right)\right) \]

   11. Applied egg-rr 73.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 10^{-60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+160}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y} \cdot z\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -10.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y) z))) (t_1 (fabs (/ x y))))
   (if (<= x -1.5e+152)
     t_0
     (if (<= x -10.2)
       t_1
       (if (<= x 8.5e-61)
         (fabs (/ 4.0 y))
         (if (<= x 0.0225) t_0 (if (<= x 2.7e+160) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x / y) * z));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.5e+152) {
		tmp = t_0;
	} else if (x <= -10.2) {
		tmp = t_1;
	} else if (x <= 8.5e-61) {
		tmp = fabs((4.0 / y));
	} else if (x <= 0.0225) {
		tmp = t_0;
	} else if (x <= 2.7e+160) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(((x / y) * z))
    t_1 = abs((x / y))
    if (x <= (-1.5d+152)) then
        tmp = t_0
    else if (x <= (-10.2d0)) then
        tmp = t_1
    else if (x <= 8.5d-61) then
        tmp = abs((4.0d0 / y))
    else if (x <= 0.0225d0) then
        tmp = t_0
    else if (x <= 2.7d+160) then
        tmp = t_1
    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) * z));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.5e+152) {
		tmp = t_0;
	} else if (x <= -10.2) {
		tmp = t_1;
	} else if (x <= 8.5e-61) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 0.0225) {
		tmp = t_0;
	} else if (x <= 2.7e+160) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x / y) * z))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.5e+152:
		tmp = t_0
	elif x <= -10.2:
		tmp = t_1
	elif x <= 8.5e-61:
		tmp = math.fabs((4.0 / y))
	elif x <= 0.0225:
		tmp = t_0
	elif x <= 2.7e+160:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x / y) * z))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.5e+152)
		tmp = t_0;
	elseif (x <= -10.2)
		tmp = t_1;
	elseif (x <= 8.5e-61)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 0.0225)
		tmp = t_0;
	elseif (x <= 2.7e+160)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x / y) * z));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.5e+152)
		tmp = t_0;
	elseif (x <= -10.2)
		tmp = t_1;
	elseif (x <= 8.5e-61)
		tmp = abs((4.0 / y));
	elseif (x <= 0.0225)
		tmp = t_0;
	elseif (x <= 2.7e+160)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.5e+152], t$95$0, If[LessEqual[x, -10.2], t$95$1, If[LessEqual[x, 8.5e-61], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0225], t$95$0, If[LessEqual[x, 2.7e+160], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999995e152 or 8.50000000000000016e-61 < x < 0.022499999999999999 or 2.7e160 < x

   1. Initial program 89.9%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\frac{{x}^{3} + {4}^{3}}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left({x}^{3} + {4}^{3}\right) \cdot \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(x \cdot z\right)\right)}{y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)}{y}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)\right), y\right)\right) \]

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \frac{x \cdot z}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{x \cdot z}{y}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \frac{x \cdot z}{y}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{x \cdot z}{y}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{z \cdot x}{y}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(z \cdot \frac{x}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 77.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right)\right) \]

   7. Simplified 77.2%

      \[\leadsto \left|\color{blue}{0 - z \cdot \frac{x}{y}}\right| \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. fabs-neg N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z}{y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right) \]

      7. *-lowering-*.f64 55.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right) \]

   9. Applied egg-rr 55.6%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right) \]

       4. /-lowering-/.f64 77.2%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   11. Applied egg-rr 77.2%

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

   if -1.49999999999999995e152 < x < -10.199999999999999 or 0.022499999999999999 < x < 2.7e160

   1. Initial program 97.2%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 71.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   7. Simplified 71.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 66.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, y\right)\right) \]

   10. Simplified 66.9%

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

   if -10.199999999999999 < x < 8.50000000000000016e-61

   1. Initial program 97.4%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 76.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   7. Simplified 76.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+152}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq -10.2:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ 4.0 y) (* (/ x y) z)))))
   (if (<= z -1.0) t_0 (if (<= z 2.6) (fabs (/ (+ 4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((4.0 / y) - ((x / y) * z)));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.6) {
		tmp = fabs(((4.0 + x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    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(((4.0d0 / y) - ((x / y) * z)))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 2.6d0) 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(((4.0 / y) - ((x / y) * z)));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.6) {
		tmp = Math.abs(((4.0 + x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((4.0 / y) - ((x / y) * z)))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 2.6:
		tmp = math.fabs(((4.0 + x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(4.0 / y) - Float64(Float64(x / y) * z)))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.6)
		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(((4.0 / y) - ((x / y) * z)));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.6)
		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[(4.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 2.6], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 2.60000000000000009 < z

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 96.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(4, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 96.7%

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

   if -1 < z < 2.60000000000000009

   1. Initial program 95.1%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \frac{x \cdot 1}{y}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + x \cdot \frac{1}{y}\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{y} \cdot \left(4 + x\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1 \cdot \left(4 + x\right)}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(4 + x\right)}{y}\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(-1 \cdot \left(4 + x\right)\right)}{y}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(4 + x\right)}{y}\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(4 + x\right)\right)}{y}\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{4 + x}{y}\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + x}{y}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 + x\right), y\right)\right) \]

      12. +-lowering-+.f64 98.3%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(4, x\right), y\right)\right) \]

   7. Simplified 98.3%

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

Alternative 5: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x \cdot z - 4}{y}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- (* x z) 4.0) y))))
   (if (<= z -1.0) t_0 (if (<= z 2.7) (fabs (/ (+ 4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((x * z) - 4.0) / y));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.7) {
		tmp = fabs(((4.0 + x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    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) - 4.0d0) / y))
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 2.7d0) 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) - 4.0) / y));
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.7) {
		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) - 4.0) / y))
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= 2.7:
		tmp = math.fabs(((4.0 + x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(x * z) - 4.0) / y))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.7)
		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) - 4.0) / y));
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.7)
		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[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 2.7], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 2.7000000000000002 < z

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 96.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(4, y\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), z\right)\right)\right) \]

   5. Simplified 96.7%

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

      1. fabs-sub N/A

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

      2. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x}{y} \cdot z - \frac{4}{y}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z}{y} - \frac{4}{y}\right)\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z - 4}{y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot z - 4\right), y\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot z\right), 4\right), y\right)\right) \]

      7. *-lowering-*.f64 91.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, z\right), 4\right), y\right)\right) \]

   7. Applied egg-rr 91.7%

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

   if -1 < z < 2.7000000000000002

   1. Initial program 95.1%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \frac{x \cdot 1}{y}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + x \cdot \frac{1}{y}\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{y} \cdot \left(4 + x\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1 \cdot \left(4 + x\right)}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(4 + x\right)}{y}\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(-1 \cdot \left(4 + x\right)\right)}{y}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(4 + x\right)}{y}\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(4 + x\right)\right)}{y}\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{4 + x}{y}\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + x}{y}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 + x\right), y\right)\right) \]

      12. +-lowering-+.f64 98.3%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(4, x\right), y\right)\right) \]

   7. Simplified 98.3%

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

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+126}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6e+126)
   (fabs (* x (/ z y)))
   (if (<= z 1.82e+52) (fabs (/ (+ 4.0 x) y)) (fabs (* (/ x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6e+126) {
		tmp = fabs((x * (z / y)));
	} else if (z <= 1.82e+52) {
		tmp = fabs(((4.0 + x) / y));
	} else {
		tmp = fabs(((x / y) * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6d+126)) then
        tmp = abs((x * (z / y)))
    else if (z <= 1.82d+52) 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 <= -6e+126) {
		tmp = Math.abs((x * (z / y)));
	} else if (z <= 1.82e+52) {
		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 <= -6e+126:
		tmp = math.fabs((x * (z / y)))
	elif z <= 1.82e+52:
		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 <= -6e+126)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (z <= 1.82e+52)
		tmp = abs(Float64(Float64(4.0 + x) / y));
	else
		tmp = abs(Float64(Float64(x / y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6e+126)
		tmp = abs((x * (z / y)));
	elseif (z <= 1.82e+52)
		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, -6e+126], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.82e+52], 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 < -6.0000000000000005e126

   1. Initial program 95.9%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\frac{{x}^{3} + {4}^{3}}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left({x}^{3} + {4}^{3}\right) \cdot \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(x \cdot z\right)\right)}{y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)}{y}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)\right), y\right)\right) \]

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \frac{x \cdot z}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{x \cdot z}{y}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \frac{x \cdot z}{y}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{x \cdot z}{y}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{z \cdot x}{y}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(z \cdot \frac{x}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 79.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right)\right) \]

   7. Simplified 79.9%

      \[\leadsto \left|\color{blue}{0 - z \cdot \frac{x}{y}}\right| \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. fabs-neg N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z}{y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right) \]

      7. *-lowering-*.f64 78.2%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right) \]

   9. Applied egg-rr 78.2%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(x \cdot \frac{z}{y}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z}{y} \cdot x\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{z}{y}\right), x\right)\right) \]

       4. /-lowering-/.f64 80.7%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), x\right)\right) \]

   11. Applied egg-rr 80.7%

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

   if -6.0000000000000005e126 < z < 1.8199999999999999e52

   1. Initial program 95.6%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + \frac{x \cdot 1}{y}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(4 \cdot \frac{1}{y} + x \cdot \frac{1}{y}\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1}{y} \cdot \left(4 + x\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{1 \cdot \left(4 + x\right)}{y}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(4 + x\right)}{y}\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(-1 \cdot \left(4 + x\right)\right)}{y}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(4 + x\right)}{y}\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(4 + x\right)\right)}{y}\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{4 + x}{y}\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4 + x}{y}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(4 + x\right), y\right)\right) \]

      12. +-lowering-+.f64 89.4%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(4, x\right), y\right)\right) \]

   7. Simplified 89.4%

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

   if 1.8199999999999999e52 < z

   1. Initial program 93.8%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

      3. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

      4. flip3-+ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\frac{{x}^{3} + {4}^{3}}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left({x}^{3} + {4}^{3}\right) \cdot \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(x \cdot z\right)\right)}{y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)}{y}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)\right), y\right)\right) \]

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(-1 \cdot \frac{x \cdot z}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\frac{x \cdot z}{y}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \frac{x \cdot z}{y}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{x \cdot z}{y}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{z \cdot x}{y}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \left(z \cdot \frac{x}{y}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 72.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right)\right) \]

   7. Simplified 72.0%

      \[\leadsto \left|\color{blue}{0 - z \cdot \frac{x}{y}}\right| \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. fabs-neg N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x \cdot z}{y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right) \]

      7. *-lowering-*.f64 64.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right) \]

   9. Applied egg-rr 64.0%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{z \cdot x}{y}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(z \cdot \frac{x}{y}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{x}{y}\right)\right)\right) \]

       4. /-lowering-/.f64 72.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(z, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   11. Applied egg-rr 72.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+126}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.82 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -10.5) t_0 (if (<= x 4.0) (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 <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    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 <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 4.0d0) 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 <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		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 <= -10.5:
		tmp = t_0
	elif x <= 4.0:
		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 <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		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 <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		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, -10.5], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 93.0%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 64.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   7. Simplified 64.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 62.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(x, y\right)\right) \]

   10. Simplified 62.4%

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

   if -10.5 < x < 4

   1. Initial program 97.7%

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

      1. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

      16. div-sub N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

      17. distribute-neg-frac N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

      18. associate--r- N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

   7. Simplified 71.0%

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

Alternative 8: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x \cdot z}{y}\right)\right) \]

   3. sub-div N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(x + 4\right) - x \cdot z}{y}\right)\right) \]

   4. flip3-+ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\frac{{x}^{3} + {4}^{3}}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

   5. div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left({x}^{3} + {4}^{3}\right) \cdot \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)} - x \cdot z}{y}\right)\right) \]

   6. fmm-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(x \cdot z\right)\right)}{y}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)}{y}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{fma}\left({x}^{3} + {4}^{3}, \frac{1}{x \cdot x + \left(4 \cdot 4 - x \cdot 4\right)}, \mathsf{neg}\left(z \cdot x\right)\right)\right), y\right)\right) \]

4. Applied egg-rr 96.2%

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

Alternative 9: 40.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \frac{x + 4}{y}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\mathsf{neg}\left(\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right)\right) \]

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right)\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\frac{x}{y} \cdot z\right) \cdot -1 - \frac{x + 4}{y} \cdot -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(-1 \cdot \left(\frac{x}{y} \cdot z\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{x + 4}{y} \cdot -1\right)\right) \]

   11. associate-*l/ N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(x + 4\right) \cdot -1}{y}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{-1 \cdot \left(x + 4\right)}{y}\right)\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\mathsf{neg}\left(\left(x + 4\right)\right)}{y}\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(4\right)\right)}{y}\right)\right) \]

   15. sub-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \frac{\left(\mathsf{neg}\left(x\right)\right) - 4}{y}\right)\right) \]

   16. div-sub N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\frac{\mathsf{neg}\left(x\right)}{y} - \frac{4}{y}\right)\right)\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right) - \frac{4}{y}\right)\right)\right) \]

   18. associate--r- N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right) + \frac{4}{y}\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{4}{y} + \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(\frac{4}{y}\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

   21. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) - \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right) \]

   22. sub-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, y\right), \left(\left(\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right)\right)\right)\right)\right) \]

3. Simplified 98.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(\frac{4}{y}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 37.9%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(4, y\right)\right) \]

7. Simplified 37.9%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))