fabs fraction 1

Percentage Accurate: 92.3% → 99.8%

Time: 12.5s

Alternatives: 12

Speedup: 1.0×

• 20.3% 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: 92.3% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) 5e+27)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (+ (/ 4.0 y) (* (/ x y) (- 1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs((((x + 4.0) / y) - ((x / y) * z))) <= 5e+27) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((4.0 / y) + ((x / y) * (1.0 - 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 (abs((((x + 4.0d0) / y) - ((x / y) * z))) <= 5d+27) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((4.0d0 / y) + ((x / y) * (1.0d0 - z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs((((x + 4.0) / y) - ((x / y) * z))) <= 5e+27) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((4.0 / y) + ((x / y) * (1.0 - z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs((((x + 4.0) / y) - ((x / y) * z))) <= 5e+27:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((4.0 / y) + ((x / y) * (1.0 - z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z))) <= 5e+27)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(4.0 / y) + Float64(Float64(x / y) * Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs((((x + 4.0) / y) - ((x / y) * z))) <= 5e+27)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((4.0 / y) + ((x / y) * (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+27], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 / y), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 4.99999999999999979e27

   1. Initial program 94.4%

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

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

   if 4.99999999999999979e27 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 90.5%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y) (- 1.0 z)))))
   (if (<= x -1.56)
     t_0
     (if (<= x 2.95e-18) (fabs (/ (- 4.0 (* x z)) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x / y) * (1.0 - z)));
	double tmp;
	if (x <= -1.56) {
		tmp = t_0;
	} else if (x <= 2.95e-18) {
		tmp = fabs(((4.0 - (x * z)) / 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) * (1.0d0 - z)))
    if (x <= (-1.56d0)) then
        tmp = t_0
    else if (x <= 2.95d-18) then
        tmp = abs(((4.0d0 - (x * z)) / 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) * (1.0 - z)));
	double tmp;
	if (x <= -1.56) {
		tmp = t_0;
	} else if (x <= 2.95e-18) {
		tmp = Math.abs(((4.0 - (x * z)) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x / y) * (1.0 - z)))
	tmp = 0
	if x <= -1.56:
		tmp = t_0
	elif x <= 2.95e-18:
		tmp = math.fabs(((4.0 - (x * z)) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x / y) * Float64(1.0 - z)))
	tmp = 0.0
	if (x <= -1.56)
		tmp = t_0;
	elseif (x <= 2.95e-18)
		tmp = abs(Float64(Float64(4.0 - Float64(x * z)) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x / y) * (1.0 - z)));
	tmp = 0.0;
	if (x <= -1.56)
		tmp = t_0;
	elseif (x <= 2.95e-18)
		tmp = abs(((4.0 - (x * z)) / y));
	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] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.56], t$95$0, If[LessEqual[x, 2.95e-18], N[Abs[N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5600000000000001 or 2.9500000000000001e-18 < x

   1. Initial program 88.1%

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

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

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

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

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      5. --lowering--.f64 92.4%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   7. Simplified 92.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \left(\frac{x}{y}\right)\right)\right) \]

      5. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   9. Applied egg-rr 98.6%

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

   if -1.5600000000000001 < x < 2.9500000000000001e-18

   1. Initial program 96.0%

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

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

   5. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{4 + \left(x - x \cdot z\right)}{y}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y) (- 1.0 z)))))
   (if (<= x -1.7e-37)
     t_0
     (if (<= x 1.2e-26) (/ 1.0 (/ 1.0 (/ (+ 4.0 (- x (* x z))) y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x / y) * (1.0 - z)));
	double tmp;
	if (x <= -1.7e-37) {
		tmp = t_0;
	} else if (x <= 1.2e-26) {
		tmp = 1.0 / (1.0 / ((4.0 + (x - (x * z))) / 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) * (1.0d0 - z)))
    if (x <= (-1.7d-37)) then
        tmp = t_0
    else if (x <= 1.2d-26) then
        tmp = 1.0d0 / (1.0d0 / ((4.0d0 + (x - (x * z))) / 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) * (1.0 - z)));
	double tmp;
	if (x <= -1.7e-37) {
		tmp = t_0;
	} else if (x <= 1.2e-26) {
		tmp = 1.0 / (1.0 / ((4.0 + (x - (x * z))) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x / y) * (1.0 - z)))
	tmp = 0
	if x <= -1.7e-37:
		tmp = t_0
	elif x <= 1.2e-26:
		tmp = 1.0 / (1.0 / ((4.0 + (x - (x * z))) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x / y) * Float64(1.0 - z)))
	tmp = 0.0
	if (x <= -1.7e-37)
		tmp = t_0;
	elseif (x <= 1.2e-26)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(4.0 + Float64(x - Float64(x * z))) / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x / y) * (1.0 - z)));
	tmp = 0.0;
	if (x <= -1.7e-37)
		tmp = t_0;
	elseif (x <= 1.2e-26)
		tmp = 1.0 / (1.0 / ((4.0 + (x - (x * z))) / y));
	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] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.7e-37], t$95$0, If[LessEqual[x, 1.2e-26], N[(1.0 / N[(1.0 / N[(N[(4.0 + N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000009e-37 or 1.2e-26 < x

   1. Initial program 88.8%

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

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

   5. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

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

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

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      5. --lowering--.f64 91.1%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   7. Simplified 91.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \left(\frac{x}{y}\right)\right)\right) \]

      5. /-lowering-/.f64 97.0%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   9. Applied egg-rr 97.0%

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

   if -1.70000000000000009e-37 < x < 1.2e-26

   1. Initial program 95.7%

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

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|\right)\right) \]

      10. fabs-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{4 + x}{y} - \frac{z}{\color{blue}{\frac{y}{x}}}\right)\right)\right) \]

      15. associate-/r/ N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

   8. Applied egg-rr 50.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{4 + \left(x - x \cdot z\right)}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 - x \cdot z}{y}\\ \mathbf{if}\;z \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- 4.0 (* x z)) y)))
   (if (<= z -8.4e+15) t_0 (if (<= z 6.3e+28) (fabs (/ (+ x 4.0) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 - (x * z)) / y;
	double tmp;
	if (z <= -8.4e+15) {
		tmp = t_0;
	} else if (z <= 6.3e+28) {
		tmp = fabs(((x + 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 = (4.0d0 - (x * z)) / y
    if (z <= (-8.4d+15)) then
        tmp = t_0
    else if (z <= 6.3d+28) then
        tmp = abs(((x + 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 = (4.0 - (x * z)) / y;
	double tmp;
	if (z <= -8.4e+15) {
		tmp = t_0;
	} else if (z <= 6.3e+28) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 - (x * z)) / y
	tmp = 0
	if z <= -8.4e+15:
		tmp = t_0
	elif z <= 6.3e+28:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 - Float64(x * z)) / y)
	tmp = 0.0
	if (z <= -8.4e+15)
		tmp = t_0;
	elseif (z <= 6.3e+28)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 - (x * z)) / y;
	tmp = 0.0;
	if (z <= -8.4e+15)
		tmp = t_0;
	elseif (z <= 6.3e+28)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -8.4e+15], t$95$0, If[LessEqual[z, 6.3e+28], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.4e15 or 6.3000000000000001e28 < z

   1. Initial program 91.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 92.1%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 92.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 92.0%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 39.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \color{blue}{\left(-1 \cdot \left(x \cdot z\right)\right)}\right), y\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(0, \left(x \cdot z\right)\right)\right), y\right) \]

       4. *-lowering-*.f64 39.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, z\right)\right)\right), y\right) \]

   11. Simplified 39.3%

       \[\leadsto \frac{4 + \color{blue}{\left(0 - x \cdot z\right)}}{y} \]

   if -8.4e15 < z < 6.3000000000000001e28

   1. Initial program 91.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.3e+16)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.3e+16) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x / y) * (1.0 - 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 (x <= 1.3d+16) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x / y) * (1.0d0 - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.3e+16) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.3e+16:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.3e+16)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.3e+16)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x / y) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.3e+16], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3e16

   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 98.0%

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

   if 1.3e16 < x

   1. Initial program 86.2%

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

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

   5. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

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

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

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      5. --lowering--.f64 92.7%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   7. Simplified 92.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \left(\frac{x}{y}\right)\right)\right) \]

      5. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, z\right), \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+155}:\\ \;\;\;\;\frac{4 + \left(x - x \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -6.6e-18)
     t_0
     (if (<= x 3e+155) (/ (+ 4.0 (- x (* x z))) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -6.6e-18) {
		tmp = t_0;
	} else if (x <= 3e+155) {
		tmp = (4.0 + (x - (x * z))) / 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 <= (-6.6d-18)) then
        tmp = t_0
    else if (x <= 3d+155) then
        tmp = (4.0d0 + (x - (x * z))) / 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 <= -6.6e-18) {
		tmp = t_0;
	} else if (x <= 3e+155) {
		tmp = (4.0 + (x - (x * z))) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -6.6e-18:
		tmp = t_0
	elif x <= 3e+155:
		tmp = (4.0 + (x - (x * z))) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -6.6e-18)
		tmp = t_0;
	elseif (x <= 3e+155)
		tmp = Float64(Float64(4.0 + Float64(x - Float64(x * z))) / 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 <= -6.6e-18)
		tmp = t_0;
	elseif (x <= 3e+155)
		tmp = (4.0 + (x - (x * z))) / 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, -6.6e-18], t$95$0, If[LessEqual[x, 3e+155], N[(N[(4.0 + N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6000000000000003e-18 or 3.0000000000000001e155 < x

   1. Initial program 84.0%

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

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

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

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

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right)\right) \]

      5. --lowering--.f64 92.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right)\right) \]

   7. Simplified 92.5%

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

   8. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 66.2%

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

   10. Simplified 66.2%

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

   if -6.6000000000000003e-18 < x < 3.0000000000000001e155

   1. Initial program 96.3%

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

      1. 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 98.2%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.1%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 46.6%

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

Alternative 7: 44.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{elif}\;x \leq 1220:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e+110)
   (* (/ x y) (- 0.0 z))
   (if (<= x 1220.0)
     (/ (- 4.0 (* x z)) y)
     (if (<= x 1.8e+156) (/ (* x (- 1.0 z)) y) (/ (+ x 4.0) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e+110) {
		tmp = (x / y) * (0.0 - z);
	} else if (x <= 1220.0) {
		tmp = (4.0 - (x * z)) / y;
	} else if (x <= 1.8e+156) {
		tmp = (x * (1.0 - z)) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	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 (x <= (-1.9d+110)) then
        tmp = (x / y) * (0.0d0 - z)
    else if (x <= 1220.0d0) then
        tmp = (4.0d0 - (x * z)) / y
    else if (x <= 1.8d+156) then
        tmp = (x * (1.0d0 - z)) / y
    else
        tmp = (x + 4.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e+110) {
		tmp = (x / y) * (0.0 - z);
	} else if (x <= 1220.0) {
		tmp = (4.0 - (x * z)) / y;
	} else if (x <= 1.8e+156) {
		tmp = (x * (1.0 - z)) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e+110:
		tmp = (x / y) * (0.0 - z)
	elif x <= 1220.0:
		tmp = (4.0 - (x * z)) / y
	elif x <= 1.8e+156:
		tmp = (x * (1.0 - z)) / y
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e+110)
		tmp = Float64(Float64(x / y) * Float64(0.0 - z));
	elseif (x <= 1220.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y);
	elseif (x <= 1.8e+156)
		tmp = Float64(Float64(x * Float64(1.0 - z)) / y);
	else
		tmp = Float64(Float64(x + 4.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.9e+110)
		tmp = (x / y) * (0.0 - z);
	elseif (x <= 1220.0)
		tmp = (4.0 - (x * z)) / y;
	elseif (x <= 1.8e+156)
		tmp = (x * (1.0 - z)) / y;
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e+110], N[(N[(x / y), $MachinePrecision] * N[(0.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1220.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.8e+156], N[(N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.89999999999999994e110

   1. Initial program 77.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.9%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.8%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 42.4%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]

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

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

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

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

       7. /-lowering-/.f64 33.2%

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

   11. Simplified 33.2%

       \[\leadsto \color{blue}{0 - z \cdot \frac{x}{y}} \]

   if -1.89999999999999994e110 < x < 1220

   1. Initial program 96.9%

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

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 98.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.7%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 46.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \color{blue}{\left(-1 \cdot \left(x \cdot z\right)\right)}\right), y\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(0, \left(x \cdot z\right)\right)\right), y\right) \]

       4. *-lowering-*.f64 42.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, z\right)\right)\right), y\right) \]

   11. Simplified 42.6%

       \[\leadsto \frac{4 + \color{blue}{\left(0 - x \cdot z\right)}}{y} \]

   if 1220 < x < 1.79999999999999989e156

   1. Initial program 96.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.9%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 91.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 91.6%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 40.8%

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

   9. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), \color{blue}{y}\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right) \]

       3. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + -1 \cdot z\right)\right), y\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right) \]

       7. --lowering--.f64 40.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right) \]

   11. Simplified 40.1%

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

   if 1.79999999999999989e156 < x

   1. Initial program 75.7%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.9%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 33.7%

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

   9. Taylor expanded in z around 0

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

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

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

       2. +-lowering-+.f64 38.4%

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

   11. Simplified 38.4%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{elif}\;x \leq 1220:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -98000000000000:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-18}:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -98000000000000.0)
   (* (/ x y) (- 0.0 z))
   (if (<= x 2.95e-18)
     (/ 4.0 y)
     (if (<= x 5.2e+155) (/ (* x (- 1.0 z)) y) (/ (+ x 4.0) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -98000000000000.0) {
		tmp = (x / y) * (0.0 - z);
	} else if (x <= 2.95e-18) {
		tmp = 4.0 / y;
	} else if (x <= 5.2e+155) {
		tmp = (x * (1.0 - z)) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	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 (x <= (-98000000000000.0d0)) then
        tmp = (x / y) * (0.0d0 - z)
    else if (x <= 2.95d-18) then
        tmp = 4.0d0 / y
    else if (x <= 5.2d+155) then
        tmp = (x * (1.0d0 - z)) / y
    else
        tmp = (x + 4.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -98000000000000.0) {
		tmp = (x / y) * (0.0 - z);
	} else if (x <= 2.95e-18) {
		tmp = 4.0 / y;
	} else if (x <= 5.2e+155) {
		tmp = (x * (1.0 - z)) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -98000000000000.0:
		tmp = (x / y) * (0.0 - z)
	elif x <= 2.95e-18:
		tmp = 4.0 / y
	elif x <= 5.2e+155:
		tmp = (x * (1.0 - z)) / y
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -98000000000000.0)
		tmp = Float64(Float64(x / y) * Float64(0.0 - z));
	elseif (x <= 2.95e-18)
		tmp = Float64(4.0 / y);
	elseif (x <= 5.2e+155)
		tmp = Float64(Float64(x * Float64(1.0 - z)) / y);
	else
		tmp = Float64(Float64(x + 4.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -98000000000000.0)
		tmp = (x / y) * (0.0 - z);
	elseif (x <= 2.95e-18)
		tmp = 4.0 / y;
	elseif (x <= 5.2e+155)
		tmp = (x * (1.0 - z)) / y;
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -98000000000000.0], N[(N[(x / y), $MachinePrecision] * N[(0.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.95e-18], N[(4.0 / y), $MachinePrecision], If[LessEqual[x, 5.2e+155], N[(N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.8e13

   1. Initial program 87.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.0%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 92.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 92.8%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 35.9%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]

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

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

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

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

       7. /-lowering-/.f64 27.2%

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

   11. Simplified 27.2%

       \[\leadsto \color{blue}{0 - z \cdot \frac{x}{y}} \]

   if -9.8e13 < x < 2.9500000000000001e-18

   1. Initial program 96.2%

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

      1. 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 100.0%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 51.1%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 36.7%

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

   11. Simplified 36.7%

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

   if 2.9500000000000001e-18 < x < 5.2000000000000004e155

   1. Initial program 97.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.4%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.1%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 38.0%

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

   9. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 - z\right)\right), \color{blue}{y}\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right) \]

       3. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + -1 \cdot z\right)\right), y\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), y\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(1 - z\right)\right), y\right) \]

       7. --lowering--.f64 37.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, z\right)\right), y\right) \]

   11. Simplified 37.0%

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

   if 5.2000000000000004e155 < x

   1. Initial program 75.7%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.9%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 33.7%

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

   9. Taylor expanded in z around 0

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

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

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

       2. +-lowering-+.f64 38.4%

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

   11. Simplified 38.4%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -98000000000000:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-18}:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot \left(1 - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -920:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{4 + \left(x - x \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -920.0)
   (* (/ x y) (- 0.0 z))
   (if (<= x 1.2e+157) (/ (+ 4.0 (- x (* x z))) y) (/ (+ x 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -920.0) {
		tmp = (x / y) * (0.0 - z);
	} else if (x <= 1.2e+157) {
		tmp = (4.0 + (x - (x * z))) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	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 (x <= (-920.0d0)) then
        tmp = (x / y) * (0.0d0 - z)
    else if (x <= 1.2d+157) then
        tmp = (4.0d0 + (x - (x * z))) / y
    else
        tmp = (x + 4.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -920.0) {
		tmp = (x / y) * (0.0 - z);
	} else if (x <= 1.2e+157) {
		tmp = (4.0 + (x - (x * z))) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -920.0:
		tmp = (x / y) * (0.0 - z)
	elif x <= 1.2e+157:
		tmp = (4.0 + (x - (x * z))) / y
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -920.0)
		tmp = Float64(Float64(x / y) * Float64(0.0 - z));
	elseif (x <= 1.2e+157)
		tmp = Float64(Float64(4.0 + Float64(x - Float64(x * z))) / y);
	else
		tmp = Float64(Float64(x + 4.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -920.0)
		tmp = (x / y) * (0.0 - z);
	elseif (x <= 1.2e+157)
		tmp = (4.0 + (x - (x * z))) / y;
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -920.0], N[(N[(x / y), $MachinePrecision] * N[(0.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+157], N[(N[(4.0 + N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -920

   1. Initial program 88.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.7%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.4%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 39.1%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]

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

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

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

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

       7. /-lowering-/.f64 26.6%

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

   11. Simplified 26.6%

       \[\leadsto \color{blue}{0 - z \cdot \frac{x}{y}} \]

   if -920 < x < 1.2e157

   1. Initial program 96.4%

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

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.1%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 47.0%

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

   if 1.2e157 < x

   1. Initial program 75.7%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.9%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 33.7%

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

   9. Taylor expanded in z around 0

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

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

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

       2. +-lowering-+.f64 38.4%

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

   11. Simplified 38.4%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -920:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{4 + \left(x - x \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.8% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0) (* (/ x y) (- 0.0 z)) (/ (+ x 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (x / y) * (0.0 - z);
	} else {
		tmp = (x + 4.0) / y;
	}
	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 (x <= (-4.0d0)) then
        tmp = (x / y) * (0.0d0 - z)
    else
        tmp = (x + 4.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (x / y) * (0.0 - z);
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = (x / y) * (0.0 - z)
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.0], N[(N[(x / y), $MachinePrecision] * N[(0.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 88.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.7%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 93.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.4%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 39.1%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{x \cdot z}{y}} \]

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

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

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

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

       7. /-lowering-/.f64 26.6%

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

   11. Simplified 26.6%

       \[\leadsto \color{blue}{0 - z \cdot \frac{x}{y}} \]

   if -4 < x

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 97.5%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

      4. div-fabs N/A

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

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

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

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

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

      7. associate--l+ N/A

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

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

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 97.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.4%

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

      1. clear-num N/A

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

      2. associate-+r- N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r/ N/A

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

      7. *-lft-identity N/A

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

      8. inv-pow N/A

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

      9. sqr-pow N/A

         \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

      10. fabs-sqr N/A

          \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

      11. sqr-pow N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-rgt-identity N/A

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

   8. Applied egg-rr 44.7%

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

   9. Taylor expanded in z around 0

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

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

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

       2. +-lowering-+.f64 35.0%

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

   11. Simplified 35.0%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x}{y} \cdot \left(0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ \frac{x + 4}{y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + 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 = (x + 4.0d0) / y
end function

Java

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

Python

def code(x, y, z):
	return (x + 4.0) / y

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 96.6%

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

   1. fabs-div N/A

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

   2. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

   4. div-fabs N/A

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

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

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

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

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

   7. associate--l+ N/A

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

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

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

   9. --lowering--.f64 N/A

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

   10. *-lowering-*.f64 96.5%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 96.5%

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

   1. clear-num N/A

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

   2. associate-+r- N/A

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

   3. div-sub N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r/ N/A

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

   7. *-lft-identity N/A

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

   8. inv-pow N/A

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

   9. sqr-pow N/A

      \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

   10. fabs-sqr N/A

       \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

   11. sqr-pow N/A

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

   12. inv-pow N/A

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

   13. clear-num N/A

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

   14. /-rgt-identity N/A

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

8. Applied egg-rr 43.4%

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

9. Taylor expanded in z around 0

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

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

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

    2. +-lowering-+.f64 33.8%

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

11. Simplified 33.8%

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

12. Final simplification 33.8%

    \[\leadsto \frac{x + 4}{y} \]
13. Add Preprocessing

Alternative 12: 21.3% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 96.6%

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

   1. fabs-div N/A

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

   2. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left|y\right|}{\left|\left(4 + x\right) - x \cdot z\right|}\right)}\right) \]

   4. div-fabs N/A

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

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

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

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

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

   7. associate--l+ N/A

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

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

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

   9. --lowering--.f64 N/A

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

   10. *-lowering-*.f64 96.5%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(4, \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, z\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 96.5%

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

   1. clear-num N/A

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

   2. associate-+r- N/A

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

   3. div-sub N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r/ N/A

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

   7. *-lft-identity N/A

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

   8. inv-pow N/A

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

   9. sqr-pow N/A

      \[\leadsto \frac{1}{\left|{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}\right|} \]

   10. fabs-sqr N/A

       \[\leadsto \frac{1}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{4 + x}{y} - 1 \cdot \frac{z}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)}}} \]

   11. sqr-pow N/A

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

   12. inv-pow N/A

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

   13. clear-num N/A

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

   14. /-rgt-identity N/A

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

8. Applied egg-rr 43.4%

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

9. Taylor expanded in x around 0

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

    1. /-lowering-/.f64 19.8%

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

11. Simplified 19.8%

    \[\leadsto \color{blue}{\frac{4}{y}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))