fabs fraction 1

Percentage Accurate: 91.6% → 99.8%

Time: 12.1s

Alternatives: 10

Speedup: 1.0×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) 2e-69)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (+ (/ 4.0 y) (* (/ x y) (- 1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs((((x + 4.0) / y) - ((x / y) * z))) <= 2e-69) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((4.0 / y) + ((x / y) * (1.0 - z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (abs((((x + 4.0d0) / y) - ((x / y) * z))) <= 2d-69) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((4.0d0 / y) + ((x / y) * (1.0d0 - z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.abs((((x + 4.0) / y) - ((x / y) * z))) <= 2e-69) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((4.0 / y) + ((x / y) * (1.0 - z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.fabs((((x + 4.0) / y) - ((x / y) * z))) <= 2e-69:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((4.0 / y) + ((x / y) * (1.0 - z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z))) <= 2e-69)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(4.0 / y) + Float64(Float64(x / y) * Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (abs((((x + 4.0) / y) - ((x / y) * z))) <= 2e-69)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((4.0 / y) + ((x / y) * (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-69], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 / y), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.9999999999999999e-69

   1. Initial program 91.1%

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

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

   if 1.9999999999999999e-69 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 93.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y) (- 1.0 z)))))
   (if (<= x -1.5) t_0 (if (<= x 0.07) (fabs (/ (- (* x z) 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 <= -1.5) {
		tmp = t_0;
	} else if (x <= 0.07) {
		tmp = fabs((((x * z) - 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 <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 0.07d0) then
        tmp = abs((((x * z) - 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 <= -1.5) {
		tmp = t_0;
	} else if (x <= 0.07) {
		tmp = Math.abs((((x * z) - 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 <= -1.5:
		tmp = t_0
	elif x <= 0.07:
		tmp = math.fabs((((x * z) - 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 <= -1.5)
		tmp = t_0;
	elseif (x <= 0.07)
		tmp = abs(Float64(Float64(Float64(x * z) - 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 <= -1.5)
		tmp = t_0;
	elseif (x <= 0.07)
		tmp = abs((((x * z) - 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, -1.5], t$95$0, If[LessEqual[x, 0.07], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 0.070000000000000007 < x

   1. Initial program 89.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 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 92.7%

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

   7. Simplified 92.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

   if -1.5 < x < 0.070000000000000007

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 95.9%

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

   5. Simplified 95.9%

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

      1. fabs-sub N/A

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

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

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

      3. associate-*l/ N/A

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

      4. sub-div N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.4%

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+77}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.45e+77)
   (fabs (* (/ x y) z))
   (if (<= z 4.2e-8) (fabs (/ (+ x 4.0) y)) (fabs (* x (/ (- 1.0 z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.45e+77) {
		tmp = fabs(((x / y) * z));
	} else if (z <= 4.2e-8) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((x * ((1.0 - 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 <= (-1.45d+77)) then
        tmp = abs(((x / y) * z))
    else if (z <= 4.2d-8) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((x * ((1.0d0 - z) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.45e+77) {
		tmp = Math.abs(((x / y) * z));
	} else if (z <= 4.2e-8) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.45e+77:
		tmp = math.fabs(((x / y) * z))
	elif z <= 4.2e-8:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.45e+77)
		tmp = abs(Float64(Float64(x / y) * z));
	elseif (z <= 4.2e-8)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.45e+77)
		tmp = abs(((x / y) * z));
	elseif (z <= 4.2e-8)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((x * ((1.0 - z) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.45e+77], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.2e-8], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4500000000000001e77

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.9%

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

   5. Simplified 98.9%

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

      1. fabs-sub N/A

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

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

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

      3. associate-*l/ N/A

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

      4. sub-div N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   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 85.3%

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

   10. Simplified 85.3%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-commutative N/A

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

       6. div-inv N/A

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

       7. /-lowering-/.f64 87.8%

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

   12. Applied egg-rr 87.8%

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

   if -1.4500000000000001e77 < z < 4.19999999999999989e-8

   1. Initial program 96.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 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 95.2%

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

   7. Simplified 95.2%

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

   if 4.19999999999999989e-8 < z

   1. Initial program 80.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 94.3%

      \[\leadsto \color{blue}{\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|} \]
   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 72.1%

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

   7. Simplified 72.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(1 - z\right)}{y}}\right| \]
   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 76.5%

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

   9. Applied egg-rr 76.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+77}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.4e+76)
   (fabs (* (/ x y) z))
   (if (<= z 2.2e-7) (fabs (/ (+ x 4.0) y)) (fabs (* (/ x y) (- 1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.4e+76) {
		tmp = fabs(((x / y) * z));
	} else if (z <= 2.2e-7) {
		tmp = fabs(((x + 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 (z <= (-4.4d+76)) then
        tmp = abs(((x / y) * z))
    else if (z <= 2.2d-7) then
        tmp = abs(((x + 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 (z <= -4.4e+76) {
		tmp = Math.abs(((x / y) * z));
	} else if (z <= 2.2e-7) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.4e+76:
		tmp = math.fabs(((x / y) * z))
	elif z <= 2.2e-7:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.4e+76)
		tmp = abs(Float64(Float64(x / y) * z));
	elseif (z <= 2.2e-7)
		tmp = abs(Float64(Float64(x + 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 (z <= -4.4e+76)
		tmp = abs(((x / y) * z));
	elseif (z <= 2.2e-7)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs(((x / y) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.4e+76], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.2e-7], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / 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 z < -4.4000000000000001e76

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.9%

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

   5. Simplified 98.9%

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

      1. fabs-sub N/A

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

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

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

      3. associate-*l/ N/A

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

      4. sub-div N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   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 85.3%

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

   10. Simplified 85.3%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-commutative N/A

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

       6. div-inv N/A

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

       7. /-lowering-/.f64 87.8%

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

   12. Applied egg-rr 87.8%

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

   if -4.4000000000000001e76 < z < 2.2000000000000001e-7

   1. Initial program 96.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 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 95.2%

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

   7. Simplified 95.2%

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

   if 2.2000000000000001e-7 < z

   1. Initial program 80.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 94.3%

      \[\leadsto \color{blue}{\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|} \]
   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 72.1%

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

   7. Simplified 72.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 75.4%

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

   9. Applied egg-rr 75.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + x \cdot \frac{1 - z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3e+78)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (+ (/ 4.0 y) (* x (/ (- 1.0 z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e+78) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((4.0 / y) + (x * ((1.0 - 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 <= 3.3d+78) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((4.0d0 / y) + (x * ((1.0d0 - z) / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e+78) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((4.0 / y) + (x * ((1.0 - z) / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3e+78:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((4.0 / y) + (x * ((1.0 - z) / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.3e+78)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(4.0 / y) + Float64(x * Float64(Float64(1.0 - z) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.3e+78)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((4.0 / y) + (x * ((1.0 - z) / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3e+78], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 / y), $MachinePrecision] + N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e78

   1. Initial program 92.9%

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

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

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

      2. associate-*l/ N/A

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

      3. sub-div N/A

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

      4. flip3-+ N/A

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

      5. div-inv N/A

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

      6. fmm-def N/A

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

      7. *-commutative N/A

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

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

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

   4. Applied egg-rr 98.6%

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

   if 3.3e78 < y

   1. Initial program 92.1%

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

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+78)
   (fabs (* (/ x y) z))
   (if (<= z 2.5e+72) (fabs (/ (+ x 4.0) y)) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+78) {
		tmp = fabs(((x / y) * z));
	} else if (z <= 2.5e+72) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((x / (y / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3d+78)) then
        tmp = abs(((x / y) * z))
    else if (z <= 2.5d+72) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+78) {
		tmp = Math.abs(((x / y) * z));
	} else if (z <= 2.5e+72) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e+78:
		tmp = math.fabs(((x / y) * z))
	elif z <= 2.5e+72:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+78)
		tmp = abs(Float64(Float64(x / y) * z));
	elseif (z <= 2.5e+72)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3e+78)
		tmp = abs(((x / y) * z));
	elseif (z <= 2.5e+72)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+78], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+72], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.99999999999999982e78

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.9%

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

   5. Simplified 98.9%

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

      1. fabs-sub N/A

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

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

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

      3. associate-*l/ N/A

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

      4. sub-div N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   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 85.3%

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

   10. Simplified 85.3%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-commutative N/A

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

       6. div-inv N/A

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

       7. /-lowering-/.f64 87.8%

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

   12. Applied egg-rr 87.8%

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

   if -2.99999999999999982e78 < z < 2.49999999999999996e72

   1. Initial program 95.8%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-frac-neg N/A

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

      10. remove-double-neg N/A

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

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

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

      12. +-lowering-+.f64 92.2%

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

   7. Simplified 92.2%

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

   if 2.49999999999999996e72 < z

   1. Initial program 78.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 93.3%

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

   5. Simplified 93.3%

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

      1. fabs-sub N/A

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

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

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

      3. associate-*l/ N/A

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

      4. sub-div N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.3%

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

   7. Applied egg-rr 91.3%

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

   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%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. /-lowering-/.f64 78.9%

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

   12. Applied egg-rr 78.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-113}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5)
   (fabs (/ x y))
   (if (<= x 2.05e-113) (fabs (/ 4.0 y)) (fabs (* (/ x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = fabs((x / y));
	} else if (x <= 2.05e-113) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = fabs(((x / y) * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.5d0)) then
        tmp = abs((x / y))
    else if (x <= 2.05d-113) then
        tmp = abs((4.0d0 / y))
    else
        tmp = abs(((x / y) * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = Math.abs((x / y));
	} else if (x <= 2.05e-113) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = Math.abs(((x / y) * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5:
		tmp = math.fabs((x / y))
	elif x <= 2.05e-113:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = math.fabs(((x / y) * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5)
		tmp = abs(Float64(x / y));
	elseif (x <= 2.05e-113)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = abs(Float64(Float64(x / y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = abs((x / y));
	elseif (x <= 2.05e-113)
		tmp = abs((4.0 / y));
	else
		tmp = abs(((x / y) * z));
	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, 2.05e-113], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 87.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 64.8%

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

   10. Simplified 64.8%

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

   if -1.5 < x < 2.05e-113

   1. Initial program 96.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 96.1%

      \[\leadsto \color{blue}{\left|\frac{4}{y} + \frac{x}{y} \cdot \left(1 - z\right)\right|} \]
   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 78.1%

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

   7. Simplified 78.1%

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

   if 2.05e-113 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 76.3%

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

   5. Simplified 76.3%

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

      1. fabs-sub N/A

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

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

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

      3. associate-*l/ N/A

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

      4. sub-div N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 68.5%

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

   7. Applied egg-rr 68.5%

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

   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 56.0%

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

   10. Simplified 56.0%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. associate-*r* N/A

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

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

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

       5. *-commutative N/A

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

       6. div-inv N/A

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

       7. /-lowering-/.f64 63.7%

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

   12. Applied egg-rr 63.7%

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

Alternative 8: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -1.55) t_0 (if (<= x 4.0) (fabs (/ 4.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -1.55:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 89.6%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 59.8%

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

   10. Simplified 59.8%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 96.4%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. neg-mul-1 N/A

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

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

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-neg-in N/A

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

      15. sub-neg N/A

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

      16. div-sub N/A

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

      17. distribute-neg-frac N/A

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

      18. associate--r- N/A

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

      19. +-commutative N/A

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

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

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

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

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

      22. sub-neg N/A

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

   3. Simplified 96.4%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 71.5%

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

   7. Simplified 71.5%

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

Alternative 9: 95.9% 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 92.8%

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

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

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

   2. associate-*l/ N/A

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

   3. sub-div N/A

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

   4. flip3-+ N/A

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

   5. div-inv N/A

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

   6. fmm-def N/A

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

   7. *-commutative N/A

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

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

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

4. Applied egg-rr 97.3%

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

5. Final simplification 97.3%

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

Alternative 10: 40.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-sub0 N/A

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

   5. associate-+l- N/A

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

   6. sub0-neg N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*l/ N/A

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

   12. *-commutative N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-neg-in N/A

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

   15. sub-neg N/A

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

   16. div-sub N/A

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

   17. distribute-neg-frac N/A

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

   18. associate--r- N/A

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

   19. +-commutative N/A

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

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

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

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

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

   22. sub-neg N/A

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 36.4%

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

7. Simplified 36.4%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))