fabs fraction 1

Percentage Accurate: 92.0% → 99.8%

Time: 13.2s

Alternatives: 14

Speedup: 1.0×

• 20.9% 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: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{y}, x + 4, 0 - \frac{x \cdot z}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) 2e-45)
   (fabs (fma (/ 1.0 y) (+ x 4.0) (- 0.0 (/ (* 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))) <= 2e-45) {
		tmp = fabs(fma((1.0 / y), (x + 4.0), (0.0 - ((x * z) / y))));
	} else {
		tmp = 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))) <= 2e-45)
		tmp = abs(fma(Float64(1.0 / y), Float64(x + 4.0), Float64(0.0 - Float64(Float64(x * z) / y))));
	else
		tmp = abs(Float64(Float64(4.0 / y) + Float64(Float64(x / y) * Float64(1.0 - z))));
	end
	return 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], 2e-45], N[Abs[N[(N[(1.0 / y), $MachinePrecision] * N[(x + 4.0), $MachinePrecision] + N[(0.0 - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $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))) < 1.99999999999999997e-45

   1. Initial program 89.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. fmm-def N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(4, x\right), \left(0 - \frac{x}{y} \cdot z\right)\right)\right) \]

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

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(4, x\right), \mathsf{\_.f64}\left(0, \left(\frac{x}{y} \cdot z\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(4, x\right), \mathsf{\_.f64}\left(0, \left(\frac{x \cdot z}{y}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(4, x\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{+.f64}\left(4, x\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{y}, 4 + x, 0 - \frac{x \cdot z}{y}\right)}\right| \]

   if 1.99999999999999997e-45 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 89.9%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{y}, x + 4, 0 - \frac{x \cdot z}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -850000:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y x)))))
   (if (<= x -3.4e+240)
     t_0
     (if (<= x -850000.0)
       (fabs (/ x y))
       (if (<= x -2.9e-108)
         (fabs (/ (* x z) y))
         (if (<= x 2e-46)
           (fabs (/ 4.0 y))
           (if (<= x 1.05e+22) t_0 (/ x y))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((z / (y / x)));
	double tmp;
	if (x <= -3.4e+240) {
		tmp = t_0;
	} else if (x <= -850000.0) {
		tmp = fabs((x / y));
	} else if (x <= -2.9e-108) {
		tmp = fabs(((x * z) / y));
	} else if (x <= 2e-46) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.05e+22) {
		tmp = t_0;
	} else {
		tmp = x / 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) :: t_0
    real(8) :: tmp
    t_0 = abs((z / (y / x)))
    if (x <= (-3.4d+240)) then
        tmp = t_0
    else if (x <= (-850000.0d0)) then
        tmp = abs((x / y))
    else if (x <= (-2.9d-108)) then
        tmp = abs(((x * z) / y))
    else if (x <= 2d-46) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.05d+22) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z / (y / x)));
	double tmp;
	if (x <= -3.4e+240) {
		tmp = t_0;
	} else if (x <= -850000.0) {
		tmp = Math.abs((x / y));
	} else if (x <= -2.9e-108) {
		tmp = Math.abs(((x * z) / y));
	} else if (x <= 2e-46) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.05e+22) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((z / (y / x)))
	tmp = 0
	if x <= -3.4e+240:
		tmp = t_0
	elif x <= -850000.0:
		tmp = math.fabs((x / y))
	elif x <= -2.9e-108:
		tmp = math.fabs(((x * z) / y))
	elif x <= 2e-46:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.05e+22:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(z / Float64(y / x)))
	tmp = 0.0
	if (x <= -3.4e+240)
		tmp = t_0;
	elseif (x <= -850000.0)
		tmp = abs(Float64(x / y));
	elseif (x <= -2.9e-108)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (x <= 2e-46)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.05e+22)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((z / (y / x)));
	tmp = 0.0;
	if (x <= -3.4e+240)
		tmp = t_0;
	elseif (x <= -850000.0)
		tmp = abs((x / y));
	elseif (x <= -2.9e-108)
		tmp = abs(((x * z) / y));
	elseif (x <= 2e-46)
		tmp = abs((4.0 / y));
	elseif (x <= 1.05e+22)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e+240], t$95$0, If[LessEqual[x, -850000.0], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -2.9e-108], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e-46], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05e+22], t$95$0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.40000000000000008e240 or 2.00000000000000005e-46 < x < 1.0499999999999999e22

   1. Initial program 86.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 79.6%

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

   5. Simplified 79.6%

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

      1. fabs-sub N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. sub-div N/A

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

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

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

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

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

      10. /-lowering-/.f64 93.4%

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

   7. Applied egg-rr 93.4%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 83.6%

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

   if -3.40000000000000008e240 < x < -8.5e5

   1. Initial program 91.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.2%

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

   7. Simplified 91.2%

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

   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 70.7%

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

   10. Simplified 70.7%

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

   if -8.5e5 < x < -2.9000000000000001e-108

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 66.0%

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

   5. Simplified 66.0%

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

      1. fabs-sub N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. sub-div N/A

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

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

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

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

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

      10. /-lowering-/.f64 65.9%

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

   7. Applied egg-rr 65.9%

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

   8. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 73.6%

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

   10. Simplified 73.6%

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

   if -2.9000000000000001e-108 < x < 2.00000000000000005e-46

   1. Initial program 95.1%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 95.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 84.0%

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

   7. Simplified 84.0%

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

   if 1.0499999999999999e22 < x

   1. Initial program 80.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.5%

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

   7. Simplified 93.5%

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

   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 76.4%

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

   10. Simplified 76.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. /-lowering-/.f64 76.3%

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

   12. Applied egg-rr 76.3%

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

       1. associate-/r/ N/A

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

       2. inv-pow N/A

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

       3. sqr-pow N/A

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

       4. fabs-sqr N/A

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

       5. sqr-pow N/A

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

       6. inv-pow N/A

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

       7. clear-num N/A

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

       8. /-lowering-/.f64 32.0%

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

   14. Applied egg-rr 32.0%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+236}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -900000:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-44}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y x)))))
   (if (<= x -8.6e+236)
     t_0
     (if (<= x -900000.0)
       (fabs (/ x y))
       (if (<= x -2.85e-35)
         t_0
         (if (<= x 8.6e-44)
           (fabs (/ 4.0 y))
           (if (<= x 7.8e+22) t_0 (/ x y))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((z / (y / x)));
	double tmp;
	if (x <= -8.6e+236) {
		tmp = t_0;
	} else if (x <= -900000.0) {
		tmp = fabs((x / y));
	} else if (x <= -2.85e-35) {
		tmp = t_0;
	} else if (x <= 8.6e-44) {
		tmp = fabs((4.0 / y));
	} else if (x <= 7.8e+22) {
		tmp = t_0;
	} else {
		tmp = x / 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) :: t_0
    real(8) :: tmp
    t_0 = abs((z / (y / x)))
    if (x <= (-8.6d+236)) then
        tmp = t_0
    else if (x <= (-900000.0d0)) then
        tmp = abs((x / y))
    else if (x <= (-2.85d-35)) then
        tmp = t_0
    else if (x <= 8.6d-44) then
        tmp = abs((4.0d0 / y))
    else if (x <= 7.8d+22) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z / (y / x)));
	double tmp;
	if (x <= -8.6e+236) {
		tmp = t_0;
	} else if (x <= -900000.0) {
		tmp = Math.abs((x / y));
	} else if (x <= -2.85e-35) {
		tmp = t_0;
	} else if (x <= 8.6e-44) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 7.8e+22) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((z / (y / x)))
	tmp = 0
	if x <= -8.6e+236:
		tmp = t_0
	elif x <= -900000.0:
		tmp = math.fabs((x / y))
	elif x <= -2.85e-35:
		tmp = t_0
	elif x <= 8.6e-44:
		tmp = math.fabs((4.0 / y))
	elif x <= 7.8e+22:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(z / Float64(y / x)))
	tmp = 0.0
	if (x <= -8.6e+236)
		tmp = t_0;
	elseif (x <= -900000.0)
		tmp = abs(Float64(x / y));
	elseif (x <= -2.85e-35)
		tmp = t_0;
	elseif (x <= 8.6e-44)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 7.8e+22)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((z / (y / x)));
	tmp = 0.0;
	if (x <= -8.6e+236)
		tmp = t_0;
	elseif (x <= -900000.0)
		tmp = abs((x / y));
	elseif (x <= -2.85e-35)
		tmp = t_0;
	elseif (x <= 8.6e-44)
		tmp = abs((4.0 / y));
	elseif (x <= 7.8e+22)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8.6e+236], t$95$0, If[LessEqual[x, -900000.0], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -2.85e-35], t$95$0, If[LessEqual[x, 8.6e-44], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 7.8e+22], t$95$0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.5999999999999997e236 or -9e5 < x < -2.8500000000000001e-35 or 8.60000000000000027e-44 < x < 7.80000000000000042e22

   1. Initial program 89.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 82.9%

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

   5. Simplified 82.9%

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

      1. fabs-sub N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. sub-div N/A

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

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

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

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

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

      10. /-lowering-/.f64 92.9%

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

   7. Applied egg-rr 92.9%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 85.6%

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

   if -8.5999999999999997e236 < x < -9e5

   1. Initial program 91.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.2%

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

   7. Simplified 91.2%

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

   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 70.7%

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

   10. Simplified 70.7%

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

   if -2.8500000000000001e-35 < x < 8.60000000000000027e-44

   1. Initial program 93.9%

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

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 80.4%

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

   7. Simplified 80.4%

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

   if 7.80000000000000042e22 < x

   1. Initial program 80.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 93.5%

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

   7. Simplified 93.5%

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

   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 76.4%

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

   10. Simplified 76.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. /-lowering-/.f64 76.3%

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

   12. Applied egg-rr 76.3%

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

       1. associate-/r/ N/A

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

       2. inv-pow N/A

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

       3. sqr-pow N/A

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

       4. fabs-sqr N/A

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

       5. sqr-pow N/A

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

       6. inv-pow N/A

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

       7. clear-num N/A

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

       8. /-lowering-/.f64 32.0%

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

   14. Applied egg-rr 32.0%

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

Alternative 4: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{if}\;x \leq -180:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x (/ y (- 1.0 z))))))
   (if (<= x -180.0) t_0 (if (<= x 4.0) (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 <= -180.0) {
		tmp = t_0;
	} else if (x <= 4.0) {
		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 <= (-180.0d0)) then
        tmp = t_0
    else if (x <= 4.0d0) 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 <= -180.0) {
		tmp = t_0;
	} else if (x <= 4.0) {
		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 <= -180.0:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs(((4.0 - (x * z)) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / Float64(y / Float64(1.0 - z))))
	tmp = 0.0
	if (x <= -180.0)
		tmp = t_0;
	elseif (x <= 4.0)
		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 <= -180.0)
		tmp = t_0;
	elseif (x <= 4.0)
		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[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -180.0], t$95$0, If[LessEqual[x, 4.0], 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 < -180 or 4 < x

   1. Initial program 85.6%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in 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 98.0%

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

   9. Applied egg-rr 98.0%

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

       1. associate-*r/ N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. associate-/r/ N/A

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

       7. remove-double-div N/A

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

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

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

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

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

       10. --lowering--.f64 98.0%

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

   11. Applied egg-rr 98.0%

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

   if -180 < x < 4

   1. Initial program 94.6%

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

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

   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 99.4%

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

Alternative 5: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x (/ y (- 1.0 z))))))
   (if (<= x -3.4e-108) t_0 (if (<= x 2.2e-44) (fabs (/ 4.0 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 <= -3.4e-108) {
		tmp = t_0;
	} else if (x <= 2.2e-44) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / (y / (1.0d0 - z))))
    if (x <= (-3.4d-108)) then
        tmp = t_0
    else if (x <= 2.2d-44) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / (y / (1.0 - z))));
	double tmp;
	if (x <= -3.4e-108) {
		tmp = t_0;
	} else if (x <= 2.2e-44) {
		tmp = Math.abs((4.0 / 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 <= -3.4e-108:
		tmp = t_0
	elif x <= 2.2e-44:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / Float64(y / Float64(1.0 - z))))
	tmp = 0.0
	if (x <= -3.4e-108)
		tmp = t_0;
	elseif (x <= 2.2e-44)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / (y / (1.0 - z))));
	tmp = 0.0;
	if (x <= -3.4e-108)
		tmp = t_0;
	elseif (x <= 2.2e-44)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e-108], t$95$0, If[LessEqual[x, 2.2e-44], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000002e-108 or 2.20000000000000012e-44 < x

   1. Initial program 86.7%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around 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 89.5%

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

   7. Simplified 89.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(1 - z\right)}{y}}\right| \]
   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 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. associate-*r/ N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. associate-/r/ N/A

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

       7. remove-double-div N/A

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

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

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

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

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

       10. --lowering--.f64 94.2%

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

   11. Applied egg-rr 94.2%

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

   if -3.40000000000000002e-108 < x < 2.20000000000000012e-44

   1. Initial program 95.1%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 95.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 84.0%

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

   7. Simplified 84.0%

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

Alternative 6: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-44}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e-108)
   (fabs (* x (/ (- 1.0 z) y)))
   (if (<= x 3.1e-44) (fabs (/ 4.0 y)) (fabs (* (/ x y) (- 1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-108) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else if (x <= 3.1e-44) {
		tmp = fabs((4.0 / 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 <= (-3.4d-108)) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else if (x <= 3.1d-44) then
        tmp = abs((4.0d0 / 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 <= -3.4e-108) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else if (x <= 3.1e-44) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e-108:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	elif x <= 3.1e-44:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e-108)
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	elseif (x <= 3.1e-44)
		tmp = abs(Float64(4.0 / 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 <= -3.4e-108)
		tmp = abs((x * ((1.0 - z) / y)));
	elseif (x <= 3.1e-44)
		tmp = abs((4.0 / y));
	else
		tmp = abs(((x / y) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e-108], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.1e-44], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000002e-108

   1. Initial program 88.9%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|} \]
   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 87.8%

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

   7. Simplified 87.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. --lowering--.f64 91.8%

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

   9. Applied egg-rr 91.8%

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

   if -3.40000000000000002e-108 < x < 3.09999999999999984e-44

   1. Initial program 95.1%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 95.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 84.0%

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

   7. Simplified 84.0%

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

   if 3.09999999999999984e-44 < x

   1. Initial program 83.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.7%

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

   7. Simplified 91.7%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(1 - z\right)}{y}}\right| \]
   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 97.1%

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

   9. Applied egg-rr 97.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-108}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-44}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y) (- 1.0 z)))))
   (if (<= x -1.9e-37) t_0 (if (<= x 1.55e-47) (fabs (/ 4.0 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 <= -1.9e-37) {
		tmp = t_0;
	} else if (x <= 1.55e-47) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x / y) * (1.0d0 - z)))
    if (x <= (-1.9d-37)) then
        tmp = t_0
    else if (x <= 1.55d-47) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((x / y) * (1.0 - z)));
	double tmp;
	if (x <= -1.9e-37) {
		tmp = t_0;
	} else if (x <= 1.55e-47) {
		tmp = Math.abs((4.0 / 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 <= -1.9e-37:
		tmp = t_0
	elif x <= 1.55e-47:
		tmp = math.fabs((4.0 / 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 <= -1.9e-37)
		tmp = t_0;
	elseif (x <= 1.55e-47)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x / y) * (1.0 - z)));
	tmp = 0.0;
	if (x <= -1.9e-37)
		tmp = t_0;
	elseif (x <= 1.55e-47)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.9e-37], t$95$0, If[LessEqual[x, 1.55e-47], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9000000000000002e-37 or 1.5499999999999999e-47 < x

   1. Initial program 87.0%

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

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

   7. Simplified 91.9%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(1 - z\right)}{y}}\right| \]
   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 -1.9000000000000002e-37 < x < 1.5499999999999999e-47

   1. Initial program 93.9%

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

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 80.4%

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

   7. Simplified 80.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+100)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+100) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x * (z / 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 (y <= 4d+100) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x * (z / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+100) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+100:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+100)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+100)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+100], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000006e100

   1. Initial program 89.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.3%

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

   if 4.00000000000000006e100 < y

   1. Initial program 91.5%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+54}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e+54)
   (fabs (/ z (/ y x)))
   (if (<= z 1.25e+19) (fabs (/ (+ x 4.0) y)) (fabs (/ (* x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e+54) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 1.25e+19) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs(((x * z) / 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 (z <= (-2.7d+54)) then
        tmp = abs((z / (y / x)))
    else if (z <= 1.25d+19) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs(((x * z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e+54) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 1.25e+19) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs(((x * z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.7e+54:
		tmp = math.fabs((z / (y / x)))
	elif z <= 1.25e+19:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs(((x * z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e+54)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 1.25e+19)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(Float64(x * z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.7e+54)
		tmp = abs((z / (y / x)));
	elseif (z <= 1.25e+19)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs(((x * z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e+54], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.25e+19], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.70000000000000011e54

   1. Initial program 94.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 73.5%

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

   5. Simplified 73.5%

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

      1. fabs-sub N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. sub-div N/A

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

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

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

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

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

      10. /-lowering-/.f64 73.5%

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

   7. Applied egg-rr 73.5%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 73.5%

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

   if -2.70000000000000011e54 < z < 1.25e19

   1. Initial program 94.3%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-frac-neg N/A

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

      10. remove-double-neg N/A

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

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

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

      12. +-lowering-+.f64 97.8%

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

   7. Simplified 97.8%

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

   if 1.25e19 < z

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 58.4%

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

   5. Simplified 58.4%

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

      1. fabs-sub N/A

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

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

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. sub-div N/A

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

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

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

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

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

      10. /-lowering-/.f64 81.0%

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

   7. Applied egg-rr 81.0%

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

   8. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 81.1%

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

   10. Simplified 81.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+54}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5) (fabs (/ x y)) (if (<= x 4.0) (fabs (/ 4.0 y)) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = fabs((x / y));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = x / 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 <= (-1.5d0)) then
        tmp = abs((x / y))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = Math.abs((x / y));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5:
		tmp = math.fabs((x / y))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5)
		tmp = abs(Float64(x / y));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = abs((x / y));
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 88.3%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 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 92.3%

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

   7. Simplified 92.3%

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

   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 66.8%

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

   10. Simplified 66.8%

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

   if -1.5 < x < 4

   1. Initial program 94.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 94.5%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 72.8%

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

   7. Simplified 72.8%

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

   if 4 < x

   1. Initial program 83.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 92.6%

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

   7. Simplified 92.6%

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

   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 70.2%

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

   10. Simplified 70.2%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. /-lowering-/.f64 70.1%

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

   12. Applied egg-rr 70.1%

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

       1. associate-/r/ N/A

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

       2. inv-pow N/A

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

       3. sqr-pow N/A

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

       4. fabs-sqr N/A

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

       5. sqr-pow N/A

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

       6. inv-pow N/A

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

       7. clear-num N/A

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

       8. /-lowering-/.f64 30.0%

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

   14. Applied egg-rr 30.0%

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

Alternative 11: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-sub0 N/A

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

   5. associate-+l- N/A

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

   6. sub0-neg N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*l/ N/A

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

   12. *-commutative N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-neg-in N/A

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

   15. sub-neg N/A

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

   16. div-sub N/A

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

   17. distribute-neg-frac N/A

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

   18. associate--r- N/A

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

   19. +-commutative N/A

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

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

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

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

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

   22. sub-neg N/A

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

3. Simplified 97.3%

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

Alternative 12: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

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

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

   2. associate-*l/ N/A

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

   3. sub-div N/A

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

   4. flip3-+ N/A

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

   5. div-inv N/A

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

   6. fmm-def N/A

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

   7. *-commutative N/A

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

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

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

4. Applied egg-rr 97.0%

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

5. Final simplification 97.0%

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

Alternative 13: 46.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 92.3%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 96.5%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 48.4%

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

   7. Simplified 48.4%

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

   if 4 < x

   1. Initial program 83.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 92.6%

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

   7. Simplified 92.6%

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

   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 70.2%

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

   10. Simplified 70.2%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. /-lowering-/.f64 70.1%

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

   12. Applied egg-rr 70.1%

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

       1. associate-/r/ N/A

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

       2. inv-pow N/A

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

       3. sqr-pow N/A

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

       4. fabs-sqr N/A

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

       5. sqr-pow N/A

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

       6. inv-pow N/A

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

       7. clear-num N/A

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

       8. /-lowering-/.f64 30.0%

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

   14. Applied egg-rr 30.0%

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

Alternative 14: 17.8% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x / 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 / y
end function

Java

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

Python

def code(x, y, z):
	return x / y

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-sub0 N/A

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

   5. associate-+l- N/A

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

   6. sub0-neg N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*l/ N/A

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

   12. *-commutative N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-neg-in N/A

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

   15. sub-neg N/A

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

   16. div-sub N/A

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

   17. distribute-neg-frac N/A

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

   18. associate--r- N/A

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

   19. +-commutative N/A

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

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

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

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

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

   22. sub-neg N/A

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

3. Simplified 97.3%

   \[\leadsto \color{blue}{\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|} \]
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 63.4%

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

7. Simplified 63.4%

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

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 38.3%

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

10. Simplified 38.3%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

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

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

    4. /-lowering-/.f64 38.3%

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

12. Applied egg-rr 38.3%

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

    1. associate-/r/ N/A

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

    2. inv-pow N/A

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

    3. sqr-pow N/A

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

    4. fabs-sqr N/A

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

    5. sqr-pow N/A

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

    6. inv-pow N/A

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

    7. clear-num N/A

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

    8. /-lowering-/.f64 18.0%

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

14. Applied egg-rr 18.0%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))