fabs fraction 1

Percentage Accurate: 91.7% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: 1.0×

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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 510000000000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 510000000000.0)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 510000000000.0) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 510000000000.0)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 510000000000.0], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1e11

   1. Initial program 89.9%

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

      1. associate-*l/ 91.4%

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

      2. sub-div 96.0%

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

   4. Applied egg-rr 96.0%

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

   if 5.1e11 < y

   1. Initial program 94.1%

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

      1. fabs-sub 94.1%

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

      2. associate-*l/ 83.1%

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

      3. associate-*r/ 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 510000000000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+57} \lor \neg \left(x \leq 1.02 \cdot 10^{+167}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -10.2)
     t_0
     (if (<= x 8.8e-31)
       (fabs (/ 4.0 y_m))
       (if (or (<= x 3.6e+57) (not (<= x 1.02e+167)))
         (fabs (* z (/ x y_m)))
         t_0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 8.8e-31) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 3.6e+57) || !(x <= 1.02e+167)) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-10.2d0)) then
        tmp = t_0
    else if (x <= 8.8d-31) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 3.6d+57) .or. (.not. (x <= 1.02d+167))) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 8.8e-31) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 3.6e+57) || !(x <= 1.02e+167)) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -10.2:
		tmp = t_0
	elif x <= 8.8e-31:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 3.6e+57) or not (x <= 1.02e+167):
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 8.8e-31)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 3.6e+57) || !(x <= 1.02e+167))
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 8.8e-31)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 3.6e+57) || ~((x <= 1.02e+167)))
		tmp = abs((z * (x / y_m)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.2], t$95$0, If[LessEqual[x, 8.8e-31], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 3.6e+57], N[Not[LessEqual[x, 1.02e+167]], $MachinePrecision]], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.199999999999999 or 3.6000000000000002e57 < x < 1.02e167

   1. Initial program 91.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. rem-square-sqrt 37.4%

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

      3. fabs-sqr 37.4%

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

      4. rem-square-sqrt 65.9%

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

      5. fabs-neg 65.9%

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

      6. distribute-neg-frac 65.9%

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

      7. distribute-neg-in 65.9%

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

      8. metadata-eval 65.9%

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

      9. +-commutative 65.9%

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

      10. sub-neg 65.9%

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

      11. rem-square-sqrt 28.2%

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

      12. fabs-sqr 28.2%

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

      13. rem-square-sqrt 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   10. Simplified 64.8%

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

   if -10.199999999999999 < x < 8.80000000000000038e-31

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 78.0%

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

   if 8.80000000000000038e-31 < x < 3.6000000000000002e57 or 1.02e167 < x

   1. Initial program 82.5%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in z around inf 54.7%

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

      1. mul-1-neg 54.7%

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

      2. distribute-frac-neg2 54.7%

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

      3. associate-/l* 69.8%

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

   7. Simplified 69.8%

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

      1. distribute-frac-neg2 69.8%

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

      2. distribute-rgt-neg-in 69.8%

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

      3. clear-num 69.7%

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

      4. div-inv 70.9%

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

      5. distribute-neg-frac 70.9%

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

   9. Applied egg-rr 70.9%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 57.9%

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

       3. sqr-neg 57.9%

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

       4. sqrt-unprod 70.6%

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

       5. add-sqr-sqrt 70.9%

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

       6. associate-/r/ 76.1%

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

   11. Applied egg-rr 76.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+57} \lor \neg \left(x \leq 1.02 \cdot 10^{+167}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-32}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+58}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -10.2)
     t_0
     (if (<= x 5.1e-32)
       (fabs (/ 4.0 y_m))
       (if (<= x 7.8e+58)
         (fabs (* x (/ z y_m)))
         (if (<= x 1.5e+161) t_0 (fabs (* z (/ x y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 5.1e-32) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 7.8e+58) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 1.5e+161) {
		tmp = t_0;
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-10.2d0)) then
        tmp = t_0
    else if (x <= 5.1d-32) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 7.8d+58) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 1.5d+161) then
        tmp = t_0
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 5.1e-32) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 7.8e+58) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 1.5e+161) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -10.2:
		tmp = t_0
	elif x <= 5.1e-32:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 7.8e+58:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 1.5e+161:
		tmp = t_0
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 5.1e-32)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 7.8e+58)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 1.5e+161)
		tmp = t_0;
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 5.1e-32)
		tmp = abs((4.0 / y_m));
	elseif (x <= 7.8e+58)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 1.5e+161)
		tmp = t_0;
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.2], t$95$0, If[LessEqual[x, 5.1e-32], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 7.8e+58], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.5e+161], t$95$0, N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -10.199999999999999 or 7.8000000000000002e58 < x < 1.50000000000000006e161

   1. Initial program 91.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. rem-square-sqrt 37.4%

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

      3. fabs-sqr 37.4%

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

      4. rem-square-sqrt 65.9%

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

      5. fabs-neg 65.9%

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

      6. distribute-neg-frac 65.9%

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

      7. distribute-neg-in 65.9%

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

      8. metadata-eval 65.9%

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

      9. +-commutative 65.9%

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

      10. sub-neg 65.9%

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

      11. rem-square-sqrt 28.2%

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

      12. fabs-sqr 28.2%

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

      13. rem-square-sqrt 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   10. Simplified 64.8%

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

   if -10.199999999999999 < x < 5.09999999999999994e-32

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 78.0%

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

   if 5.09999999999999994e-32 < x < 7.8000000000000002e58

   1. Initial program 91.9%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. distribute-frac-neg2 68.4%

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

      3. associate-/l* 68.8%

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

   7. Simplified 68.8%

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

      1. distribute-frac-neg2 68.8%

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

      2. distribute-rgt-neg-in 68.8%

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

      3. clear-num 68.6%

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

      4. div-inv 71.6%

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

      5. distribute-neg-frac 71.6%

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

   9. Applied egg-rr 71.6%

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

       1. clear-num 71.6%

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

       2. associate-/r/ 68.6%

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

       3. clear-num 68.8%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 68.8%

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

       6. sqr-neg 68.8%

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

       7. sqrt-unprod 68.4%

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

       8. add-sqr-sqrt 68.8%

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

   11. Applied egg-rr 68.8%

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

   if 1.50000000000000006e161 < x

   1. Initial program 75.6%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in z around inf 44.8%

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

      1. mul-1-neg 44.8%

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

      2. distribute-frac-neg2 44.8%

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

      3. associate-/l* 70.6%

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

   7. Simplified 70.6%

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

      1. distribute-frac-neg2 70.6%

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

      2. distribute-rgt-neg-in 70.6%

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

      3. clear-num 70.4%

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

      4. div-inv 70.3%

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

      5. distribute-neg-frac 70.3%

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

   9. Applied egg-rr 70.3%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 47.9%

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

       3. sqr-neg 47.9%

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

       4. sqrt-unprod 70.1%

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

       5. add-sqr-sqrt 70.3%

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

       6. associate-/r/ 81.5%

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

   11. Applied egg-rr 81.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-32}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+58}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -10.2)
     t_0
     (if (<= x 2.55e-33)
       (fabs (/ 4.0 y_m))
       (if (<= x 2.25e+57)
         (fabs (/ x (/ y_m z)))
         (if (<= x 4.7e+162) t_0 (fabs (* z (/ x y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 2.55e-33) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 2.25e+57) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= 4.7e+162) {
		tmp = t_0;
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-10.2d0)) then
        tmp = t_0
    else if (x <= 2.55d-33) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 2.25d+57) then
        tmp = abs((x / (y_m / z)))
    else if (x <= 4.7d+162) then
        tmp = t_0
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 2.55e-33) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 2.25e+57) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= 4.7e+162) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -10.2:
		tmp = t_0
	elif x <= 2.55e-33:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 2.25e+57:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= 4.7e+162:
		tmp = t_0
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 2.55e-33)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 2.25e+57)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= 4.7e+162)
		tmp = t_0;
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 2.55e-33)
		tmp = abs((4.0 / y_m));
	elseif (x <= 2.25e+57)
		tmp = abs((x / (y_m / z)));
	elseif (x <= 4.7e+162)
		tmp = t_0;
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.2], t$95$0, If[LessEqual[x, 2.55e-33], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.25e+57], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.7e+162], t$95$0, N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -10.199999999999999 or 2.24999999999999998e57 < x < 4.70000000000000003e162

   1. Initial program 91.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. rem-square-sqrt 37.4%

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

      3. fabs-sqr 37.4%

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

      4. rem-square-sqrt 65.9%

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

      5. fabs-neg 65.9%

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

      6. distribute-neg-frac 65.9%

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

      7. distribute-neg-in 65.9%

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

      8. metadata-eval 65.9%

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

      9. +-commutative 65.9%

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

      10. sub-neg 65.9%

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

      11. rem-square-sqrt 28.2%

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

      12. fabs-sqr 28.2%

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

      13. rem-square-sqrt 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   10. Simplified 64.8%

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

   if -10.199999999999999 < x < 2.55000000000000004e-33

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 78.0%

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

   if 2.55000000000000004e-33 < x < 2.24999999999999998e57

   1. Initial program 91.9%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

      3. distribute-lft-neg-in 68.4%

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

      4. *-commutative 68.4%

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

      5. associate-*r/ 68.6%

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

   7. Simplified 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/r/ 71.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 71.6%

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

      5. sqr-neg 71.6%

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

      6. sqrt-unprod 71.3%

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

      7. add-sqr-sqrt 71.6%

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

   9. Applied egg-rr 71.6%

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

   if 4.70000000000000003e162 < x

   1. Initial program 75.6%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in z around inf 44.8%

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

      1. mul-1-neg 44.8%

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

      2. distribute-frac-neg2 44.8%

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

      3. associate-/l* 70.6%

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

   7. Simplified 70.6%

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

      1. distribute-frac-neg2 70.6%

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

      2. distribute-rgt-neg-in 70.6%

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

      3. clear-num 70.4%

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

      4. div-inv 70.3%

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

      5. distribute-neg-frac 70.3%

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

   9. Applied egg-rr 70.3%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 47.9%

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

       3. sqr-neg 47.9%

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

       4. sqrt-unprod 70.1%

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

       5. add-sqr-sqrt 70.3%

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

       6. associate-/r/ 81.5%

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

   11. Applied egg-rr 81.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+162}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -10.5)
     t_0
     (if (<= x 4.8e-31)
       (fabs (/ 4.0 y_m))
       (if (<= x 1.6e+57)
         (fabs (/ x (/ y_m z)))
         (if (<= x 1.1e+166) t_0 (fabs (/ z (/ y_m x)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.8e-31) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 1.6e+57) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= 1.1e+166) {
		tmp = t_0;
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 4.8d-31) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 1.6d+57) then
        tmp = abs((x / (y_m / z)))
    else if (x <= 1.1d+166) then
        tmp = t_0
    else
        tmp = abs((z / (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.8e-31) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 1.6e+57) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= 1.1e+166) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -10.5:
		tmp = t_0
	elif x <= 4.8e-31:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 1.6e+57:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= 1.1e+166:
		tmp = t_0
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.8e-31)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 1.6e+57)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= 1.1e+166)
		tmp = t_0;
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.8e-31)
		tmp = abs((4.0 / y_m));
	elseif (x <= 1.6e+57)
		tmp = abs((x / (y_m / z)));
	elseif (x <= 1.1e+166)
		tmp = t_0;
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 4.8e-31], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.6e+57], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1e+166], t$95$0, N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -10.5 or 1.60000000000000015e57 < x < 1.1e166

   1. Initial program 91.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. rem-square-sqrt 37.4%

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

      3. fabs-sqr 37.4%

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

      4. rem-square-sqrt 65.9%

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

      5. fabs-neg 65.9%

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

      6. distribute-neg-frac 65.9%

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

      7. distribute-neg-in 65.9%

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

      8. metadata-eval 65.9%

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

      9. +-commutative 65.9%

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

      10. sub-neg 65.9%

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

      11. rem-square-sqrt 28.2%

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

      12. fabs-sqr 28.2%

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

      13. rem-square-sqrt 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in x around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   10. Simplified 64.8%

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

   if -10.5 < x < 4.8e-31

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 78.0%

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

   if 4.8e-31 < x < 1.60000000000000015e57

   1. Initial program 91.9%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

      3. distribute-lft-neg-in 68.4%

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

      4. *-commutative 68.4%

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

      5. associate-*r/ 68.6%

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

   7. Simplified 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/r/ 71.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 71.6%

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

      5. sqr-neg 71.6%

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

      6. sqrt-unprod 71.3%

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

      7. add-sqr-sqrt 71.6%

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

   9. Applied egg-rr 71.6%

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

   if 1.1e166 < x

   1. Initial program 75.6%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in z around inf 44.8%

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

      1. associate-*r/ 44.8%

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

      2. neg-mul-1 44.8%

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

      3. distribute-lft-neg-in 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-*r/ 81.5%

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

   7. Simplified 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-/r/ 70.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 47.9%

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

      5. sqr-neg 47.9%

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

      6. sqrt-unprod 70.1%

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

      7. add-sqr-sqrt 70.3%

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

      8. associate-/r/ 81.5%

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

      9. clear-num 81.4%

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

      10. associate-*l/ 81.6%

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

      11. *-un-lft-identity 81.6%

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

   9. Applied egg-rr 81.6%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+166}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -2200000 \lor \neg \left(z \leq 3.35\right):\\ \;\;\;\;\left|z \cdot \frac{x - \frac{4}{z}}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -2200000.0) (not (<= z 3.35)))
   (fabs (* z (/ (- x (/ 4.0 z)) y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -2200000.0) || !(z <= 3.35)) {
		tmp = fabs((z * ((x - (4.0 / z)) / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2200000.0d0)) .or. (.not. (z <= 3.35d0))) then
        tmp = abs((z * ((x - (4.0d0 / z)) / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -2200000.0) || !(z <= 3.35)) {
		tmp = Math.abs((z * ((x - (4.0 / z)) / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -2200000.0) or not (z <= 3.35):
		tmp = math.fabs((z * ((x - (4.0 / z)) / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -2200000.0) || !(z <= 3.35))
		tmp = abs(Float64(z * Float64(Float64(x - Float64(4.0 / z)) / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -2200000.0) || ~((z <= 3.35)))
		tmp = abs((z * ((x - (4.0 / z)) / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -2200000.0], N[Not[LessEqual[z, 3.35]], $MachinePrecision]], N[Abs[N[(z * N[(N[(x - N[(4.0 / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e6 or 3.35000000000000009 < z

   1. Initial program 90.3%

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

      1. fabs-sub 90.3%

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

      2. associate-*l/ 81.8%

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

      3. associate-*r/ 89.4%

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

      4. fmm-def 91.7%

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

      5. distribute-neg-frac 91.7%

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

      6. +-commutative 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. unsub-neg 91.7%

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

      9. metadata-eval 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in z around inf 89.3%

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

      1. +-commutative 89.3%

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

      2. mul-1-neg 89.3%

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

      3. mul-1-neg 89.3%

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

      4. associate-*r/ 89.3%

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

      5. distribute-lft-in 89.3%

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

      6. metadata-eval 89.3%

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

      7. neg-mul-1 89.3%

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

      8. sub-neg 89.3%

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

      9. associate-/r* 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in y around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. unsub-neg 91.6%

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

   10. Simplified 91.6%

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

   11. Taylor expanded in x around 0 90.6%

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

   if -2.2e6 < z < 3.35000000000000009

   1. Initial program 91.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. rem-square-sqrt 51.7%

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

      3. fabs-sqr 51.7%

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

      4. rem-square-sqrt 99.7%

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

      5. fabs-neg 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

      10. sub-neg 99.7%

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

      11. rem-square-sqrt 47.4%

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

      12. fabs-sqr 47.4%

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

      13. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2200000 \lor \neg \left(z \leq 3.35\right):\\ \;\;\;\;\left|z \cdot \frac{x - \frac{4}{z}}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+133} \lor \neg \left(x \leq 2.25 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.6e+133) (not (<= x 2.25e+16)))
   (fabs (* (/ x y_m) (+ z -1.0)))
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.6e+133) || !(x <= 2.25e+16)) {
		tmp = fabs(((x / y_m) * (z + -1.0)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.6d+133)) .or. (.not. (x <= 2.25d+16))) then
        tmp = abs(((x / y_m) * (z + (-1.0d0))))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.6e+133) || !(x <= 2.25e+16)) {
		tmp = Math.abs(((x / y_m) * (z + -1.0)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.6e+133) or not (x <= 2.25e+16):
		tmp = math.fabs(((x / y_m) * (z + -1.0)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.6e+133) || !(x <= 2.25e+16))
		tmp = abs(Float64(Float64(x / y_m) * Float64(z + -1.0)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.6e+133) || ~((x <= 2.25e+16)))
		tmp = abs(((x / y_m) * (z + -1.0)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.6e+133], N[Not[LessEqual[x, 2.25e+16]], $MachinePrecision]], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999999e133 or 2.25e16 < x

   1. Initial program 83.3%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around inf 81.3%

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

      1. mul-1-neg 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

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

      6. associate-+l- 99.9%

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

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -1.59999999999999999e133 < x < 2.25e16

   1. Initial program 95.5%

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

      1. associate-*l/ 99.9%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+133} \lor \neg \left(x \leq 2.25 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 8.2e+37)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8.2e+37) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 8.2d+37) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8.2e+37) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 8.2e+37:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 8.2e+37)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 8.2e+37)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 8.2e+37], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.1999999999999996e37

   1. Initial program 90.1%

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

      1. associate-*l/ 91.6%

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

      2. sub-div 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 8.1999999999999996e37 < y

   1. Initial program 93.5%

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

      1. associate-*l/ 81.6%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -125000000:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -125000000.0)
   (fabs (/ x (/ y_m z)))
   (if (<= z 4.2e+27) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -125000000.0) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 4.2e+27) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-125000000.0d0)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 4.2d+27) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -125000000.0) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 4.2e+27) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -125000000.0:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 4.2e+27:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -125000000.0)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 4.2e+27)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -125000000.0)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 4.2e+27)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -125000000.0], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.2e+27], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e8

   1. Initial program 94.2%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in z around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

      3. distribute-lft-neg-in 66.4%

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

      4. *-commutative 66.4%

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

      5. associate-*r/ 70.7%

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

   7. Simplified 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-/r/ 72.1%

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

      3. add-sqr-sqrt 34.0%

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

      4. sqrt-unprod 59.5%

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

      5. sqr-neg 59.5%

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

      6. sqrt-unprod 37.9%

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

      7. add-sqr-sqrt 72.1%

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

   9. Applied egg-rr 72.1%

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

   if -1.25e8 < z < 4.19999999999999989e27

   1. Initial program 91.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. rem-square-sqrt 51.2%

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

      3. fabs-sqr 51.2%

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

      4. rem-square-sqrt 99.0%

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

      5. fabs-neg 99.0%

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

      6. distribute-neg-frac 99.0%

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

      7. distribute-neg-in 99.0%

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

      8. metadata-eval 99.0%

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

      9. +-commutative 99.0%

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

      10. sub-neg 99.0%

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

      11. rem-square-sqrt 47.3%

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

      12. fabs-sqr 47.3%

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

      13. rem-square-sqrt 99.0%

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

   7. Simplified 99.0%

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

   if 4.19999999999999989e27 < z

   1. Initial program 86.6%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in z around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. distribute-frac-neg2 64.3%

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

      3. associate-/l* 80.1%

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

   7. Simplified 80.1%

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

      1. distribute-frac-neg2 80.1%

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

      2. distribute-rgt-neg-in 80.1%

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

      3. clear-num 79.9%

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

      4. div-inv 80.1%

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

      5. distribute-neg-frac 80.1%

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

   9. Applied egg-rr 80.1%

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

       1. add-sqr-sqrt 37.5%

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

       2. sqrt-unprod 52.6%

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

       3. sqr-neg 52.6%

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

       4. sqrt-unprod 42.0%

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

       5. add-sqr-sqrt 80.1%

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

       6. associate-/r/ 80.6%

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

   11. Applied egg-rr 80.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -125000000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -11500000:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(z + -1\right)\right|\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -11500000.0)
   (fabs (* (/ x y_m) (+ z -1.0)))
   (if (<= z 7.2e+27) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -11500000.0) {
		tmp = fabs(((x / y_m) * (z + -1.0)));
	} else if (z <= 7.2e+27) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-11500000.0d0)) then
        tmp = abs(((x / y_m) * (z + (-1.0d0))))
    else if (z <= 7.2d+27) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -11500000.0) {
		tmp = Math.abs(((x / y_m) * (z + -1.0)));
	} else if (z <= 7.2e+27) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -11500000.0:
		tmp = math.fabs(((x / y_m) * (z + -1.0)))
	elif z <= 7.2e+27:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -11500000.0)
		tmp = abs(Float64(Float64(x / y_m) * Float64(z + -1.0)));
	elseif (z <= 7.2e+27)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -11500000.0)
		tmp = abs(((x / y_m) * (z + -1.0)));
	elseif (z <= 7.2e+27)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -11500000.0], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 7.2e+27], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e7

   1. Initial program 94.2%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in x around inf 67.9%

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

      1. mul-1-neg 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-/l* 72.2%

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

      4. distribute-lft-neg-in 72.2%

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

      5. neg-sub0 72.2%

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

      6. associate-+l- 72.2%

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

      7. neg-sub0 72.2%

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

      8. +-commutative 72.2%

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

      9. unsub-neg 72.2%

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

   7. Simplified 72.2%

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

   if -1.15e7 < z < 7.19999999999999966e27

   1. Initial program 91.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. rem-square-sqrt 51.2%

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

      3. fabs-sqr 51.2%

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

      4. rem-square-sqrt 99.0%

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

      5. fabs-neg 99.0%

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

      6. distribute-neg-frac 99.0%

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

      7. distribute-neg-in 99.0%

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

      8. metadata-eval 99.0%

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

      9. +-commutative 99.0%

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

      10. sub-neg 99.0%

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

      11. rem-square-sqrt 47.3%

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

      12. fabs-sqr 47.3%

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

      13. rem-square-sqrt 99.0%

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

   7. Simplified 99.0%

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

   if 7.19999999999999966e27 < z

   1. Initial program 86.6%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in z around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. distribute-frac-neg2 64.3%

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

      3. associate-/l* 80.1%

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

   7. Simplified 80.1%

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

      1. distribute-frac-neg2 80.1%

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

      2. distribute-rgt-neg-in 80.1%

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

      3. clear-num 79.9%

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

      4. div-inv 80.1%

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

      5. distribute-neg-frac 80.1%

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

   9. Applied egg-rr 80.1%

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

       1. add-sqr-sqrt 37.5%

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

       2. sqrt-unprod 52.6%

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

       3. sqr-neg 52.6%

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

       4. sqrt-unprod 42.0%

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

       5. add-sqr-sqrt 80.1%

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

       6. associate-/r/ 80.6%

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

   11. Applied egg-rr 80.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -11500000:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.5% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 88.1%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around 0 59.7%

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

      1. +-commutative 59.7%

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

      2. rem-square-sqrt 34.9%

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

      3. fabs-sqr 34.9%

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

      4. rem-square-sqrt 59.7%

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

      5. fabs-neg 59.7%

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

      6. distribute-neg-frac 59.7%

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

      7. distribute-neg-in 59.7%

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

      8. metadata-eval 59.7%

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

      9. +-commutative 59.7%

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

      10. sub-neg 59.7%

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

      11. rem-square-sqrt 24.6%

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

      12. fabs-sqr 24.6%

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

      13. rem-square-sqrt 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in x around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. neg-mul-1 58.1%

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

   10. Simplified 58.1%

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

   if -10.5 < x < 4

   1. Initial program 94.0%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.3%

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

4. Final simplification 66.2%

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

Alternative 12: 40.7% accurate, 1.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ 4.0 y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Simplified 92.7%

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

5. Taylor expanded in x around 0 38.1%

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

6. Final simplification 38.1%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))