fabs fraction 1

Percentage Accurate: 91.6% → 98.0%

Time: 5.5s

Alternatives: 9

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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+47)
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+47) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2d+47)) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+47) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e+47:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+47)
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e+47)
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e+47], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e47

   1. Initial program 73.9%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-lft-in 77.7%

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

      5. associate-*r/ 77.8%

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

      6. *-rgt-identity 77.8%

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

      7. mul-1-neg 77.8%

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

      8. distribute-rgt-neg-in 77.8%

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

      9. unsub-neg 77.8%

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

      10. *-lft-identity 77.8%

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

      11. associate-/l* 77.8%

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

      12. *-commutative 77.8%

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

      13. associate-/r/ 73.8%

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

      14. div-sub 99.8%

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

      15. associate-/r/ 99.8%

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

   4. Simplified 99.8%

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

   if -2.0000000000000001e47 < x

   1. Initial program 95.3%

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

      1. associate-*l/ 97.0%

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

      2. sub-div 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 2: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -6.5e+140)
     t_0
     (if (<= x -9e+39)
       t_1
       (if (<= x -6.8e-16)
         t_0
         (if (<= x 1.55e-53)
           (fabs (/ 4.0 y))
           (if (<= x 1.65e+134) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -6.5e+140) {
		tmp = t_0;
	} else if (x <= -9e+39) {
		tmp = t_1;
	} else if (x <= -6.8e-16) {
		tmp = t_0;
	} else if (x <= 1.55e-53) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.65e+134) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-6.5d+140)) then
        tmp = t_0
    else if (x <= (-9d+39)) then
        tmp = t_1
    else if (x <= (-6.8d-16)) then
        tmp = t_0
    else if (x <= 1.55d-53) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.65d+134) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -6.5e+140) {
		tmp = t_0;
	} else if (x <= -9e+39) {
		tmp = t_1;
	} else if (x <= -6.8e-16) {
		tmp = t_0;
	} else if (x <= 1.55e-53) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.65e+134) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -6.5e+140:
		tmp = t_0
	elif x <= -9e+39:
		tmp = t_1
	elif x <= -6.8e-16:
		tmp = t_0
	elif x <= 1.55e-53:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.65e+134:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -6.5e+140)
		tmp = t_0;
	elseif (x <= -9e+39)
		tmp = t_1;
	elseif (x <= -6.8e-16)
		tmp = t_0;
	elseif (x <= 1.55e-53)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.65e+134)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -6.5e+140)
		tmp = t_0;
	elseif (x <= -9e+39)
		tmp = t_1;
	elseif (x <= -6.8e-16)
		tmp = t_0;
	elseif (x <= 1.55e-53)
		tmp = abs((4.0 / y));
	elseif (x <= 1.65e+134)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e+140], t$95$0, If[LessEqual[x, -9e+39], t$95$1, If[LessEqual[x, -6.8e-16], t$95$0, If[LessEqual[x, 1.55e-53], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.65e+134], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4999999999999999e140 or -8.99999999999999991e39 < x < -6.8e-16 or 1.55000000000000008e-53 < x < 1.65e134

   1. Initial program 86.5%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in z around inf 60.4%

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

      1. add-sqr-sqrt 29.1%

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

      2. sqrt-unprod 47.4%

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

      3. sqr-neg 47.4%

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

      4. sqrt-unprod 31.2%

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

      5. add-sqr-sqrt 60.4%

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

      6. *-commutative 60.4%

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

      7. associate-*l/ 70.4%

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

      8. clear-num 70.4%

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

      9. clear-num 70.4%

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

      10. add-sqr-sqrt 37.9%

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

      11. sqrt-unprod 52.0%

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

      12. sqr-neg 52.0%

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

      13. sqrt-unprod 32.4%

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

      14. add-sqr-sqrt 70.4%

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

   6. Applied egg-rr 70.4%

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

   if -6.4999999999999999e140 < x < -8.99999999999999991e39 or 1.65e134 < x

   1. Initial program 85.2%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 88.6%

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

      5. associate-*r/ 88.7%

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

      6. *-rgt-identity 88.7%

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

      7. mul-1-neg 88.7%

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

      8. distribute-rgt-neg-in 88.7%

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

      9. unsub-neg 88.7%

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

      10. *-lft-identity 88.7%

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

      11. associate-/l* 88.6%

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

      12. *-commutative 88.6%

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

      13. associate-/r/ 86.8%

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

      14. div-sub 99.7%

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

      15. associate-/r/ 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in z around 0 74.4%

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

   if -6.8e-16 < x < 1.55000000000000008e-53

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 86.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+140}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+39}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-16}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+134}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+135}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y x)))) (t_1 (fabs (/ x y))))
   (if (<= x -1.65e+135)
     (fabs (* z (/ x y)))
     (if (<= x -9e+39)
       t_1
       (if (<= x -6e-15)
         t_0
         (if (<= x 6.2e-47)
           (fabs (/ 4.0 y))
           (if (<= x 1.75e+130) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((z / (y / x)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.65e+135) {
		tmp = fabs((z * (x / y)));
	} else if (x <= -9e+39) {
		tmp = t_1;
	} else if (x <= -6e-15) {
		tmp = t_0;
	} else if (x <= 6.2e-47) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.75e+130) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z / (y / x)))
    t_1 = abs((x / y))
    if (x <= (-1.65d+135)) then
        tmp = abs((z * (x / y)))
    else if (x <= (-9d+39)) then
        tmp = t_1
    else if (x <= (-6d-15)) then
        tmp = t_0
    else if (x <= 6.2d-47) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.75d+130) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z / (y / x)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.65e+135) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= -9e+39) {
		tmp = t_1;
	} else if (x <= -6e-15) {
		tmp = t_0;
	} else if (x <= 6.2e-47) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.75e+130) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((z / (y / x)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.65e+135:
		tmp = math.fabs((z * (x / y)))
	elif x <= -9e+39:
		tmp = t_1
	elif x <= -6e-15:
		tmp = t_0
	elif x <= 6.2e-47:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.75e+130:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(z / Float64(y / x)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.65e+135)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= -9e+39)
		tmp = t_1;
	elseif (x <= -6e-15)
		tmp = t_0;
	elseif (x <= 6.2e-47)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.75e+130)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((z / (y / x)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.65e+135)
		tmp = abs((z * (x / y)));
	elseif (x <= -9e+39)
		tmp = t_1;
	elseif (x <= -6e-15)
		tmp = t_0;
	elseif (x <= 6.2e-47)
		tmp = abs((4.0 / y));
	elseif (x <= 1.75e+130)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65e+135], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -9e+39], t$95$1, If[LessEqual[x, -6e-15], t$95$0, If[LessEqual[x, 6.2e-47], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.75e+130], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.65e135

   1. Initial program 64.4%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in z around inf 54.8%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-unprod 53.4%

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

      3. sqr-neg 53.4%

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

      4. sqrt-unprod 34.3%

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

      5. add-sqr-sqrt 54.8%

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

      6. *-commutative 54.8%

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

      7. associate-*l/ 81.6%

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

      8. clear-num 81.5%

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

      9. clear-num 81.6%

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

      10. add-sqr-sqrt 52.1%

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

      11. sqrt-unprod 65.7%

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

      12. sqr-neg 65.7%

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

      13. sqrt-unprod 29.4%

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

      14. add-sqr-sqrt 81.6%

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

   6. Applied egg-rr 81.6%

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

   if -1.65e135 < x < -8.99999999999999991e39 or 1.75e130 < x

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 88.9%

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

      5. associate-*r/ 89.1%

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

      6. *-rgt-identity 89.1%

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

      7. mul-1-neg 89.1%

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

      8. distribute-rgt-neg-in 89.1%

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

      9. unsub-neg 89.1%

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

      10. *-lft-identity 89.1%

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

      11. associate-/l* 89.0%

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

      12. *-commutative 89.0%

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

      13. associate-/r/ 87.2%

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

      14. div-sub 99.7%

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

      15. associate-/r/ 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in z around 0 73.6%

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

   if -8.99999999999999991e39 < x < -6e-15 or 6.1999999999999996e-47 < x < 1.75e130

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. mul-1-neg 64.2%

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

      3. distribute-rgt-neg-out 64.2%

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

      4. associate-*r/ 64.2%

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

      5. distribute-frac-neg 64.2%

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

      6. mul-1-neg 64.2%

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

      7. metadata-eval 64.2%

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

      8. times-frac 64.2%

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

      9. *-lft-identity 64.2%

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

      10. neg-mul-1 64.2%

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

   4. Simplified 64.2%

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

      1. associate-*r/ 64.2%

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

      2. add-sqr-sqrt 28.4%

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

      3. sqrt-unprod 45.5%

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

      4. sqr-neg 45.5%

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

      5. sqrt-unprod 35.6%

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

      6. add-sqr-sqrt 64.2%

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

      7. associate-/l* 64.3%

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

   6. Applied egg-rr 64.3%

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

   if -6e-15 < x < 6.1999999999999996e-47

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 86.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+135}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+39}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+130}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y)))))
   (if (<= x -5e-17)
     t_0
     (if (<= x 5.8e-47)
       (fabs (/ 4.0 y))
       (if (<= x 2.5e+129) t_0 (fabs (/ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x * (z / y)));
	double tmp;
	if (x <= -5e-17) {
		tmp = t_0;
	} else if (x <= 5.8e-47) {
		tmp = fabs((4.0 / y));
	} else if (x <= 2.5e+129) {
		tmp = t_0;
	} else {
		tmp = fabs((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    if (x <= (-5d-17)) then
        tmp = t_0
    else if (x <= 5.8d-47) then
        tmp = abs((4.0d0 / y))
    else if (x <= 2.5d+129) then
        tmp = t_0
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -5e-17) {
		tmp = t_0;
	} else if (x <= 5.8e-47) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 2.5e+129) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -5e-17:
		tmp = t_0
	elif x <= 5.8e-47:
		tmp = math.fabs((4.0 / y))
	elif x <= 2.5e+129:
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -5e-17)
		tmp = t_0;
	elseif (x <= 5.8e-47)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 2.5e+129)
		tmp = t_0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -5e-17)
		tmp = t_0;
	elseif (x <= 5.8e-47)
		tmp = abs((4.0 / y));
	elseif (x <= 2.5e+129)
		tmp = t_0;
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5e-17], t$95$0, If[LessEqual[x, 5.8e-47], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.5e+129], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999999e-17 or 5.8000000000000001e-47 < x < 2.5000000000000001e129

   1. Initial program 87.3%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in z around inf 54.2%

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

      1. associate-*l/ 61.4%

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

      2. *-commutative 61.4%

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

   6. Simplified 61.4%

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

   if -4.9999999999999999e-17 < x < 5.8000000000000001e-47

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 86.8%

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

   if 2.5000000000000001e129 < x

   1. Initial program 81.8%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 87.6%

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

      5. associate-*r/ 87.8%

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

      6. *-rgt-identity 87.8%

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

      7. mul-1-neg 87.8%

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

      8. distribute-rgt-neg-in 87.8%

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

      9. unsub-neg 87.8%

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

      10. *-lft-identity 87.8%

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

      11. associate-/l* 87.5%

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

      12. *-commutative 87.5%

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

      13. associate-/r/ 84.5%

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

      14. div-sub 99.7%

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

      15. associate-/r/ 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in z around 0 77.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+129}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-15} \lor \neg \left(x \leq 2.5 \cdot 10^{-53}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.1e-15) (not (<= x 2.5e-53)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e-15) || !(x <= 2.5e-53)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.1d-15)) .or. (.not. (x <= 2.5d-53))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e-15) || !(x <= 2.5e-53)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.1e-15) or not (x <= 2.5e-53):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.1e-15) || !(x <= 2.5e-53))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.1e-15) || ~((x <= 2.5e-53)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.1e-15], N[Not[LessEqual[x, 2.5e-53]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999981e-15 or 2.5e-53 < x

   1. Initial program 86.0%

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

   2. Taylor expanded in x around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. sub-neg 93.3%

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

      3. mul-1-neg 93.3%

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

      4. distribute-lft-in 82.3%

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

      5. associate-*r/ 82.5%

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

      6. *-rgt-identity 82.5%

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

      7. mul-1-neg 82.5%

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

      8. distribute-rgt-neg-in 82.5%

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

      9. unsub-neg 82.5%

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

      10. *-lft-identity 82.5%

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

      11. associate-/l* 82.4%

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

      12. *-commutative 82.4%

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

      13. associate-/r/ 80.8%

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

      14. div-sub 94.0%

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

      15. associate-/r/ 93.3%

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

   4. Simplified 93.3%

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

   if -2.09999999999999981e-15 < x < 2.5e-53

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 86.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-15} \lor \neg \left(x \leq 2.5 \cdot 10^{-53}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 6: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-13} \lor \neg \left(x \leq 4.8 \cdot 10^{-49}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e-13) (not (<= x 4.8e-49)))
   (fabs (/ (+ z -1.0) (/ y x)))
   (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-13) || !(x <= 4.8e-49)) {
		tmp = fabs(((z + -1.0) / (y / x)));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.9d-13)) .or. (.not. (x <= 4.8d-49))) then
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-13) || !(x <= 4.8e-49)) {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e-13) or not (x <= 4.8e-49):
		tmp = math.fabs(((z + -1.0) / (y / x)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e-13) || !(x <= 4.8e-49))
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.9e-13) || ~((x <= 4.8e-49)))
		tmp = abs(((z + -1.0) / (y / x)));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e-13], N[Not[LessEqual[x, 4.8e-49]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999998e-13 or 4.79999999999999985e-49 < x

   1. Initial program 86.0%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around inf 86.6%

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

      1. associate-/l* 94.0%

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

      2. sub-neg 94.0%

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

      3. metadata-eval 94.0%

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

   6. Simplified 94.0%

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

   if -2.8999999999999998e-13 < x < 4.79999999999999985e-49

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 86.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-13} \lor \neg \left(x \leq 4.8 \cdot 10^{-49}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+39}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.9e+39)
   (fabs (* z (/ x y)))
   (if (<= z 3.7e+93) (fabs (/ (- -4.0 x) y)) (fabs (/ (* x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.9e+39) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 3.7e+93) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((x * z) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.9d+39)) then
        tmp = abs((z * (x / y)))
    else if (z <= 3.7d+93) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((x * z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.9e+39) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 3.7e+93) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((x * z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.9e+39:
		tmp = math.fabs((z * (x / y)))
	elif z <= 3.7e+93:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((x * z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.9e+39)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 3.7e+93)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(x * z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.9e+39)
		tmp = abs((z * (x / y)));
	elseif (z <= 3.7e+93)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((x * z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.9e+39], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.7e+93], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.89999999999999987e39

   1. Initial program 97.9%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in z around inf 62.1%

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

      1. add-sqr-sqrt 25.8%

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

      2. sqrt-unprod 42.6%

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

      3. sqr-neg 42.6%

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

      4. sqrt-unprod 36.1%

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

      5. add-sqr-sqrt 62.1%

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

      6. *-commutative 62.1%

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

      7. associate-*l/ 73.7%

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

      8. clear-num 73.7%

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

      9. clear-num 73.7%

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

      10. add-sqr-sqrt 41.9%

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

      11. sqrt-unprod 47.1%

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

      12. sqr-neg 47.1%

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

      13. sqrt-unprod 31.6%

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

      14. add-sqr-sqrt 73.7%

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

   6. Applied egg-rr 73.7%

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

   if -4.89999999999999987e39 < z < 3.69999999999999987e93

   1. Initial program 92.4%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 93.0%

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

      1. associate-*r/ 93.0%

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

      2. distribute-lft-in 93.0%

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

      3. metadata-eval 93.0%

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

      4. neg-mul-1 93.0%

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

      5. sub-neg 93.0%

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

   6. Simplified 93.0%

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

   if 3.69999999999999987e93 < z

   1. Initial program 72.5%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in z around inf 75.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+39}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+93}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \end{array} \]

Alternative 8: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 84.3%

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

   2. Taylor expanded in x around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. sub-neg 98.2%

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

      3. mul-1-neg 98.2%

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

      4. distribute-lft-in 85.9%

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

      5. associate-*r/ 86.1%

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

      6. *-rgt-identity 86.1%

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

      7. mul-1-neg 86.1%

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

      8. distribute-rgt-neg-in 86.1%

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

      9. unsub-neg 86.1%

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

      10. *-lft-identity 86.1%

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

      11. associate-/l* 86.0%

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

      12. *-commutative 86.0%

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

      13. associate-/r/ 83.5%

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

      14. div-sub 98.3%

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

      15. associate-/r/ 98.2%

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

   4. Simplified 98.2%

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

   5. Taylor expanded in z around 0 60.9%

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

   if -10.5 < x < 4

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 80.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 9: 40.2% 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 91.1%

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

2. Taylor expanded in x around 0 44.5%

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

3. Final simplification 44.5%

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

Reproduce

herbie shell --seed 2023192 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))