fabs fraction 1

Percentage Accurate: 91.7% → 96.9%

Time: 6.9s

Alternatives: 7

Speedup: 1.6×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+18)
   (fabs (- (/ 4.0 y) (* z (/ x y))))
   (fabs (/ (- x (fma x z -4.0)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+18) {
		tmp = fabs(((4.0 / y) - (z * (x / y))));
	} else {
		tmp = fabs(((x - fma(x, z, -4.0)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+18)
		tmp = abs(Float64(Float64(4.0 / y) - Float64(z * Float64(x / y))));
	else
		tmp = abs(Float64(Float64(x - fma(x, z, -4.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+18], N[Abs[N[(N[(4.0 / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - N[(x * z + -4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2e18

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 99.8

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

   5. Simplified 99.8%

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

   if -2e18 < z

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-/l* N/A

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

      9. mul-1-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   5. Simplified 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{if}\;z \leq -280000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma x z -4.0) y))))
   (if (<= z -280000.0) t_0 (if (<= z 6.5e-34) (fabs (/ (+ 4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((fma(x, z, -4.0) / y));
	double tmp;
	if (z <= -280000.0) {
		tmp = t_0;
	} else if (z <= 6.5e-34) {
		tmp = fabs(((4.0 + x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(fma(x, z, -4.0) / y))
	tmp = 0.0
	if (z <= -280000.0)
		tmp = t_0;
	elseif (z <= 6.5e-34)
		tmp = abs(Float64(Float64(4.0 + x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -280000.0], t$95$0, If[LessEqual[z, 6.5e-34], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e5 or 6.49999999999999985e-34 < z

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 95.5

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

   5. Simplified 95.5%

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

   6. Taylor expanded in y around 0

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

      1. fabs-sub N/A

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

      2. lower-fabs.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval 93.1

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

   8. Simplified 93.1%

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

   if -2.8e5 < z < 6.49999999999999985e-34

   1. Initial program 93.3%

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

   3. Taylor expanded in z around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -280000:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{z \cdot x}{y}\right|\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+47}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* z x) y))))
   (if (<= z -1.05e+50) t_0 (if (<= z 5.1e+47) (fabs (/ (+ 4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((z * x) / y));
	double tmp;
	if (z <= -1.05e+50) {
		tmp = t_0;
	} else if (z <= 5.1e+47) {
		tmp = fabs(((4.0 + x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((z * x) / y))
    if (z <= (-1.05d+50)) then
        tmp = t_0
    else if (z <= 5.1d+47) then
        tmp = abs(((4.0d0 + x) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((z * x) / y));
	double tmp;
	if (z <= -1.05e+50) {
		tmp = t_0;
	} else if (z <= 5.1e+47) {
		tmp = Math.abs(((4.0 + x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((z * x) / y))
	tmp = 0
	if z <= -1.05e+50:
		tmp = t_0
	elif z <= 5.1e+47:
		tmp = math.fabs(((4.0 + x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(z * x) / y))
	tmp = 0.0
	if (z <= -1.05e+50)
		tmp = t_0;
	elseif (z <= 5.1e+47)
		tmp = abs(Float64(Float64(4.0 + x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((z * x) / y));
	tmp = 0.0;
	if (z <= -1.05e+50)
		tmp = t_0;
	elseif (z <= 5.1e+47)
		tmp = abs(((4.0 + x) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.05e+50], t$95$0, If[LessEqual[z, 5.1e+47], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e50 or 5.1000000000000001e47 < z

   1. Initial program 89.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. fabs-neg N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 76.7

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

   8. Simplified 76.7%

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

   if -1.05e50 < z < 5.1000000000000001e47

   1. Initial program 92.8%

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

   3. Taylor expanded in z around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 95.0

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

   5. Simplified 95.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+47}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -10.0) t_0 (if (<= x 4.0) (fabs (/ 4.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -10.0) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if (x <= (-10.0d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -10.0) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -10.0:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -10.0)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -10.0)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.0], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -10 or 4 < x

   1. Initial program 87.7%

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

   3. Taylor expanded in z around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. neg-mul-1 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 68.7

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

   5. Simplified 68.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 67.1

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

   8. Simplified 67.1%

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

   if -10 < x < 4

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 78.7

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

   5. Simplified 78.7%

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

Alternative 5: 96.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. associate-/l* N/A

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

   9. mul-1-neg N/A

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

   10. associate-+r+ N/A

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

   11. +-commutative N/A

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

   12. associate-*r/ N/A

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

   13. *-rgt-identity N/A

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

   14. +-commutative N/A

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

   15. associate-+r+ N/A

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

5. Simplified 97.0%

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

Alternative 6: 69.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in z around 0

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. neg-mul-1 N/A

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

   10. remove-double-neg N/A

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

   11. lower-/.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 73.8

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

5. Simplified 73.8%

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

6. Final simplification 73.8%

   \[\leadsto \left|\frac{4 + x}{y}\right| \]
7. Add Preprocessing

Alternative 7: 40.3% accurate, 2.6× 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 91.5%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 38.6

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

5. Simplified 38.6%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))