fabs fraction 1

Percentage Accurate: 92.1% → 99.8%

Time: 7.7s

Alternatives: 11

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: 92.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e+35)
   (fabs (/ (- (+ 4.0 x) (* 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 <= 2e+35) {
		tmp = fabs((((4.0 + x) - (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 <= 2e+35)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - 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, 2e+35], N[Abs[N[(N[(N[(4.0 + x), $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 < 1.9999999999999999e35

   1. Initial program 91.9%

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

   2. Taylor expanded in y around 0 96.6%

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

   if 1.9999999999999999e35 < y

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{4 + x}{y_m} - z \cdot \frac{x}{y_m}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ 4.0 x) y_m) (* z (/ x y_m)))))
   (if (<= t_0 -2e+82) (fabs t_0) (fabs (/ (- (+ 4.0 x) (* x z)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((4.0 + x) / y_m) - (z * (x / y_m));
	double tmp;
	if (t_0 <= -2e+82) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((((4.0 + x) - (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) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 + x) / y_m) - (z * (x / y_m))
    if (t_0 <= (-2d+82)) then
        tmp = abs(t_0)
    else
        tmp = abs((((4.0d0 + x) - (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 t_0 = ((4.0 + x) / y_m) - (z * (x / y_m));
	double tmp;
	if (t_0 <= -2e+82) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = ((4.0 + x) / y_m) - (z * (x / y_m))
	tmp = 0
	if t_0 <= -2e+82:
		tmp = math.fabs(t_0)
	else:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(4.0 + x) / y_m) - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_0 <= -2e+82)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = ((4.0 + x) / y_m) - (z * (x / y_m));
	tmp = 0.0;
	if (t_0 <= -2e+82)
		tmp = abs(t_0);
	else
		tmp = abs((((4.0 + x) - (x * z)) / 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[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+82], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < -1.9999999999999999e82

   1. Initial program 99.9%

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

   if -1.9999999999999999e82 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 89.5%

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

   2. Taylor expanded in y around 0 96.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - z \cdot \frac{x}{y} \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 3: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+208} \lor \neg \left(x \leq 1.9 \cdot 10^{+295}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -8.6e+189)
     t_0
     (if (<= x -8e+34)
       t_1
       (if (<= x -5.8e-35)
         t_0
         (if (<= x 1.75e-25)
           (fabs (/ 4.0 y_m))
           (if (or (<= x 7.2e+208) (not (<= x 1.9e+295))) t_0 t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -8.6e+189) {
		tmp = t_0;
	} else if (x <= -8e+34) {
		tmp = t_1;
	} else if (x <= -5.8e-35) {
		tmp = t_0;
	} else if (x <= 1.75e-25) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 7.2e+208) || !(x <= 1.9e+295)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-8.6d+189)) then
        tmp = t_0
    else if (x <= (-8d+34)) then
        tmp = t_1
    else if (x <= (-5.8d-35)) then
        tmp = t_0
    else if (x <= 1.75d-25) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 7.2d+208) .or. (.not. (x <= 1.9d+295))) then
        tmp = t_0
    else
        tmp = t_1
    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((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -8.6e+189) {
		tmp = t_0;
	} else if (x <= -8e+34) {
		tmp = t_1;
	} else if (x <= -5.8e-35) {
		tmp = t_0;
	} else if (x <= 1.75e-25) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 7.2e+208) || !(x <= 1.9e+295)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -8.6e+189:
		tmp = t_0
	elif x <= -8e+34:
		tmp = t_1
	elif x <= -5.8e-35:
		tmp = t_0
	elif x <= 1.75e-25:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 7.2e+208) or not (x <= 1.9e+295):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -8.6e+189)
		tmp = t_0;
	elseif (x <= -8e+34)
		tmp = t_1;
	elseif (x <= -5.8e-35)
		tmp = t_0;
	elseif (x <= 1.75e-25)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 7.2e+208) || !(x <= 1.9e+295))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -8.6e+189)
		tmp = t_0;
	elseif (x <= -8e+34)
		tmp = t_1;
	elseif (x <= -5.8e-35)
		tmp = t_0;
	elseif (x <= 1.75e-25)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 7.2e+208) || ~((x <= 1.9e+295)))
		tmp = t_0;
	else
		tmp = t_1;
	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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8.6e+189], t$95$0, If[LessEqual[x, -8e+34], t$95$1, If[LessEqual[x, -5.8e-35], t$95$0, If[LessEqual[x, 1.75e-25], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 7.2e+208], N[Not[LessEqual[x, 1.9e+295]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.59999999999999995e189 or -7.99999999999999956e34 < x < -5.8000000000000004e-35 or 1.7500000000000001e-25 < x < 7.20000000000000005e208 or 1.9000000000000001e295 < x

   1. Initial program 92.1%

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

   2. Taylor expanded in z around inf 47.3%

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

      1. mul-1-neg 47.3%

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

      2. associate-*l/ 68.5%

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

      3. distribute-rgt-neg-out 68.5%

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

   4. Simplified 68.5%

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

      1. add-sqr-sqrt 36.4%

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

      2. sqrt-unprod 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 32.0%

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

      5. add-sqr-sqrt 68.5%

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

      6. associate-/r/ 61.2%

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

   6. Applied egg-rr 61.2%

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

      1. associate-/r/ 68.5%

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

   8. Applied egg-rr 68.5%

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

   if -8.59999999999999995e189 < x < -7.99999999999999956e34 or 7.20000000000000005e208 < x < 1.9000000000000001e295

   1. Initial program 85.7%

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

   2. Taylor expanded in z around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. metadata-eval 77.9%

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

   4. Simplified 77.9%

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

   5. Taylor expanded in x around inf 77.9%

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

   if -5.8000000000000004e-35 < x < 1.7500000000000001e-25

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 81.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+189}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-35}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+208} \lor \neg \left(x \leq 1.9 \cdot 10^{+295}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+296}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -4.4e+191)
     t_0
     (if (<= x -3e+34)
       t_1
       (if (<= x -7.8e-36)
         t_0
         (if (<= x 6.5e-25)
           (fabs (/ 4.0 y_m))
           (if (<= x 1.55e+209)
             t_0
             (if (<= x 1.1e+296) t_1 (fabs (/ z (/ y_m x)))))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -4.4e+191) {
		tmp = t_0;
	} else if (x <= -3e+34) {
		tmp = t_1;
	} else if (x <= -7.8e-36) {
		tmp = t_0;
	} else if (x <= 6.5e-25) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 1.55e+209) {
		tmp = t_0;
	} else if (x <= 1.1e+296) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-4.4d+191)) then
        tmp = t_0
    else if (x <= (-3d+34)) then
        tmp = t_1
    else if (x <= (-7.8d-36)) then
        tmp = t_0
    else if (x <= 6.5d-25) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 1.55d+209) then
        tmp = t_0
    else if (x <= 1.1d+296) then
        tmp = t_1
    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((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -4.4e+191) {
		tmp = t_0;
	} else if (x <= -3e+34) {
		tmp = t_1;
	} else if (x <= -7.8e-36) {
		tmp = t_0;
	} else if (x <= 6.5e-25) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 1.55e+209) {
		tmp = t_0;
	} else if (x <= 1.1e+296) {
		tmp = t_1;
	} 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((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -4.4e+191:
		tmp = t_0
	elif x <= -3e+34:
		tmp = t_1
	elif x <= -7.8e-36:
		tmp = t_0
	elif x <= 6.5e-25:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 1.55e+209:
		tmp = t_0
	elif x <= 1.1e+296:
		tmp = t_1
	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(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -4.4e+191)
		tmp = t_0;
	elseif (x <= -3e+34)
		tmp = t_1;
	elseif (x <= -7.8e-36)
		tmp = t_0;
	elseif (x <= 6.5e-25)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 1.55e+209)
		tmp = t_0;
	elseif (x <= 1.1e+296)
		tmp = t_1;
	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((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -4.4e+191)
		tmp = t_0;
	elseif (x <= -3e+34)
		tmp = t_1;
	elseif (x <= -7.8e-36)
		tmp = t_0;
	elseif (x <= 6.5e-25)
		tmp = abs((4.0 / y_m));
	elseif (x <= 1.55e+209)
		tmp = t_0;
	elseif (x <= 1.1e+296)
		tmp = t_1;
	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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.4e+191], t$95$0, If[LessEqual[x, -3e+34], t$95$1, If[LessEqual[x, -7.8e-36], t$95$0, If[LessEqual[x, 6.5e-25], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.55e+209], t$95$0, If[LessEqual[x, 1.1e+296], t$95$1, N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.4e191 or -3.00000000000000018e34 < x < -7.8000000000000001e-36 or 6.5e-25 < x < 1.55e209

   1. Initial program 92.0%

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

   2. Taylor expanded in z around inf 48.3%

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

      1. mul-1-neg 48.3%

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

      2. associate-*l/ 67.7%

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

      3. distribute-rgt-neg-out 67.7%

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

   4. Simplified 67.7%

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

      1. add-sqr-sqrt 36.0%

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

      2. sqrt-unprod 46.7%

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

      3. sqr-neg 46.7%

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

      4. sqrt-unprod 31.6%

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

      5. add-sqr-sqrt 67.7%

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

      6. associate-/r/ 61.5%

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

   6. Applied egg-rr 61.5%

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

      1. associate-/r/ 67.7%

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

   8. Applied egg-rr 67.7%

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

   if -4.4e191 < x < -3.00000000000000018e34 or 1.55e209 < x < 1.10000000000000007e296

   1. Initial program 85.7%

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

   2. Taylor expanded in z around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. metadata-eval 77.9%

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

   4. Simplified 77.9%

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

   5. Taylor expanded in x around inf 77.9%

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

   if -7.8000000000000001e-36 < x < 6.5e-25

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 81.4%

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

   if 1.10000000000000007e296 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in z around inf 9.5%

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

      1. mul-1-neg 9.5%

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

      2. associate-*l/ 99.2%

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

      3. distribute-rgt-neg-out 99.2%

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

   4. Simplified 99.2%

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

      1. add-sqr-sqrt 50.0%

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

      2. sqrt-unprod 55.9%

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

      3. sqr-neg 55.9%

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

      4. sqrt-unprod 49.2%

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

      5. add-sqr-sqrt 99.2%

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

      6. associate-*l/ 9.5%

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

      7. expm1-log1p-u 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      8. expm1-udef 0.0%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]

      9. associate-*l/ 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{y} \cdot z}\right)} - 1\right| \]

      10. clear-num 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot z\right)} - 1\right| \]

      11. associate-*l/ 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot z}{\frac{y}{x}}}\right)} - 1\right| \]

      12. *-un-lft-identity 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{z}}{\frac{y}{x}}\right)} - 1\right| \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{\frac{y}{x}}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{\frac{y}{x}}\right)\right)}\right| \]

      2. expm1-log1p 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+191}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-36}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+209}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+296}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 5: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* x (/ z y_m)))))
   (if (<= x -3.05e+34)
     t_0
     (if (<= x -4.8e-36)
       t_1
       (if (<= x 3.1e-25) (fabs (/ 4.0 y_m)) (if (<= x 5.5e+155) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -3.05e+34) {
		tmp = t_0;
	} else if (x <= -4.8e-36) {
		tmp = t_1;
	} else if (x <= 3.1e-25) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 5.5e+155) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((x * (z / y_m)))
    if (x <= (-3.05d+34)) then
        tmp = t_0
    else if (x <= (-4.8d-36)) then
        tmp = t_1
    else if (x <= 3.1d-25) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 5.5d+155) then
        tmp = t_1
    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 t_1 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -3.05e+34) {
		tmp = t_0;
	} else if (x <= -4.8e-36) {
		tmp = t_1;
	} else if (x <= 3.1e-25) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 5.5e+155) {
		tmp = t_1;
	} 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))
	t_1 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -3.05e+34:
		tmp = t_0
	elif x <= -4.8e-36:
		tmp = t_1
	elif x <= 3.1e-25:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 5.5e+155:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -3.05e+34)
		tmp = t_0;
	elseif (x <= -4.8e-36)
		tmp = t_1;
	elseif (x <= 3.1e-25)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 5.5e+155)
		tmp = t_1;
	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));
	t_1 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -3.05e+34)
		tmp = t_0;
	elseif (x <= -4.8e-36)
		tmp = t_1;
	elseif (x <= 3.1e-25)
		tmp = abs((4.0 / y_m));
	elseif (x <= 5.5e+155)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.05e+34], t$95$0, If[LessEqual[x, -4.8e-36], t$95$1, If[LessEqual[x, 3.1e-25], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.5e+155], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.04999999999999998e34 or 5.5000000000000001e155 < x

   1. Initial program 87.0%

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

   2. Taylor expanded in z around 0 70.8%

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

      1. associate-*r/ 70.8%

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

      2. metadata-eval 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in x around inf 70.8%

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

   if -3.04999999999999998e34 < x < -4.8e-36 or 3.09999999999999995e-25 < x < 5.5000000000000001e155

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 53.6%

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

      1. mul-1-neg 53.6%

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

      2. associate-*l/ 62.5%

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

      3. distribute-rgt-neg-out 62.5%

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

   4. Simplified 62.5%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 46.6%

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

      3. sqr-neg 46.6%

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

      4. sqrt-unprod 28.2%

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

      5. add-sqr-sqrt 62.5%

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

      6. associate-*l/ 53.6%

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

      7. expm1-log1p-u 22.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      8. expm1-udef 15.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]

      9. associate-*l/ 15.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{y} \cdot z}\right)} - 1\right| \]

      10. clear-num 15.3%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot z\right)} - 1\right| \]

      11. associate-*l/ 15.3%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot z}{\frac{y}{x}}}\right)} - 1\right| \]

      12. *-un-lft-identity 15.3%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{z}}{\frac{y}{x}}\right)} - 1\right| \]

   6. Applied egg-rr 15.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{\frac{y}{x}}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 22.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{\frac{y}{x}}\right)\right)}\right| \]

      2. expm1-log1p 60.1%

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

      3. associate-/r/ 60.2%

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

      4. *-commutative 60.2%

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

   8. Simplified 60.2%

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

   if -4.8e-36 < x < 3.09999999999999995e-25

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 81.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-36}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+155}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+16} \lor \neg \left(x \leq 7.2 \cdot 10^{+15}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -2.4e+16) (not (<= x 7.2e+15)))
   (fabs (/ x (/ y_m (- 1.0 z))))
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -2.4e+16) || !(x <= 7.2e+15)) {
		tmp = fabs((x / (y_m / (1.0 - z))));
	} else {
		tmp = fabs((((4.0 + x) - (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 <= (-2.4d+16)) .or. (.not. (x <= 7.2d+15))) then
        tmp = abs((x / (y_m / (1.0d0 - z))))
    else
        tmp = abs((((4.0d0 + x) - (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 <= -2.4e+16) || !(x <= 7.2e+15)) {
		tmp = Math.abs((x / (y_m / (1.0 - z))));
	} else {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -2.4e+16) or not (x <= 7.2e+15):
		tmp = math.fabs((x / (y_m / (1.0 - z))))
	else:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -2.4e+16) || !(x <= 7.2e+15))
		tmp = abs(Float64(x / Float64(y_m / Float64(1.0 - z))));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) - 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 <= -2.4e+16) || ~((x <= 7.2e+15)))
		tmp = abs((x / (y_m / (1.0 - z))));
	else
		tmp = abs((((4.0 + x) - (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, -2.4e+16], N[Not[LessEqual[x, 7.2e+15]], $MachinePrecision]], N[Abs[N[(x / N[(y$95$m / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e16 or 7.2e15 < x

   1. Initial program 88.0%

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

   2. Taylor expanded in y around 0 90.5%

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

   3. Taylor expanded in x around inf 90.5%

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

      1. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -2.4e16 < x < 7.2e15

   1. Initial program 96.3%

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

   2. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+16} \lor \neg \left(x \leq 7.2 \cdot 10^{+15}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 7: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-35} \lor \neg \left(x \leq 6.2 \cdot 10^{-25}\right):\\ \;\;\;\;\left|x \cdot \frac{z + -1}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.5e-35) (not (<= x 6.2e-25)))
   (fabs (* x (/ (+ z -1.0) 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 ((x <= -1.5e-35) || !(x <= 6.2e-25)) {
		tmp = fabs((x * ((z + -1.0) / 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 ((x <= (-1.5d-35)) .or. (.not. (x <= 6.2d-25))) then
        tmp = abs((x * ((z + (-1.0d0)) / 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 ((x <= -1.5e-35) || !(x <= 6.2e-25)) {
		tmp = Math.abs((x * ((z + -1.0) / 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 (x <= -1.5e-35) or not (x <= 6.2e-25):
		tmp = math.fabs((x * ((z + -1.0) / 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 ((x <= -1.5e-35) || !(x <= 6.2e-25))
		tmp = abs(Float64(x * Float64(Float64(z + -1.0) / 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 ((x <= -1.5e-35) || ~((x <= 6.2e-25)))
		tmp = abs((x * ((z + -1.0) / 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[x, -1.5e-35], N[Not[LessEqual[x, 6.2e-25]], $MachinePrecision]], N[Abs[N[(x * N[(N[(z + -1.0), $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 x < -1.49999999999999994e-35 or 6.19999999999999989e-25 < x

   1. Initial program 89.5%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 95.4%

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

   5. Taylor expanded in y around 0 95.4%

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

   if -1.49999999999999994e-35 < x < 6.19999999999999989e-25

   1. Initial program 95.8%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around 0 81.4%

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

      1. associate-*r/ 81.4%

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

      2. distribute-lft-in 81.4%

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

      3. metadata-eval 81.4%

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

      4. neg-mul-1 81.4%

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

      5. sub-neg 81.4%

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

   6. Simplified 81.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-35} \lor \neg \left(x \leq 6.2 \cdot 10^{-25}\right):\\ \;\;\;\;\left|x \cdot \frac{z + -1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 8: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-35} \lor \neg \left(x \leq 8.5 \cdot 10^{-26}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -3.8e-35) (not (<= x 8.5e-26)))
   (fabs (/ x (/ y_m (- 1.0 z))))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -3.8e-35) || !(x <= 8.5e-26)) {
		tmp = fabs((x / (y_m / (1.0 - z))));
	} 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 ((x <= (-3.8d-35)) .or. (.not. (x <= 8.5d-26))) then
        tmp = abs((x / (y_m / (1.0d0 - z))))
    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 ((x <= -3.8e-35) || !(x <= 8.5e-26)) {
		tmp = Math.abs((x / (y_m / (1.0 - z))));
	} 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 (x <= -3.8e-35) or not (x <= 8.5e-26):
		tmp = math.fabs((x / (y_m / (1.0 - z))))
	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 ((x <= -3.8e-35) || !(x <= 8.5e-26))
		tmp = abs(Float64(x / Float64(y_m / Float64(1.0 - z))));
	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 ((x <= -3.8e-35) || ~((x <= 8.5e-26)))
		tmp = abs((x / (y_m / (1.0 - z))));
	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[x, -3.8e-35], N[Not[LessEqual[x, 8.5e-26]], $MachinePrecision]], N[Abs[N[(x / N[(y$95$m / N[(1.0 - z), $MachinePrecision]), $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 x < -3.8000000000000001e-35 or 8.50000000000000004e-26 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in y around 0 91.7%

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

   3. Taylor expanded in x around inf 87.4%

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

      1. associate-/l* 95.6%

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

   5. Simplified 95.6%

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

   if -3.8000000000000001e-35 < x < 8.50000000000000004e-26

   1. Initial program 95.8%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around 0 81.4%

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

      1. associate-*r/ 81.4%

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

      2. distribute-lft-in 81.4%

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

      3. metadata-eval 81.4%

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

      4. neg-mul-1 81.4%

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

      5. sub-neg 81.4%

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

   6. Simplified 81.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-35} \lor \neg \left(x \leq 8.5 \cdot 10^{-26}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 9: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+66}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -3.9e+66)
   (fabs (* z (/ x y_m)))
   (if (<= z 1.45e+155) (fabs (/ (- -4.0 x) y_m)) (fabs (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -3.9e+66) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 1.45e+155) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((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 (z <= (-3.9d+66)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 1.45d+155) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((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 (z <= -3.9e+66) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 1.45e+155) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -3.9e+66:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 1.45e+155:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -3.9e+66)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 1.45e+155)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -3.9e+66)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 1.45e+155)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * (z / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -3.9e+66], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.45e+155], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9000000000000004e66

   1. Initial program 97.7%

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

   2. Taylor expanded in z around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. associate-*l/ 82.6%

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

      3. distribute-rgt-neg-out 82.6%

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

   4. Simplified 82.6%

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

      1. add-sqr-sqrt 82.3%

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

      2. sqrt-unprod 53.8%

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

      3. sqr-neg 53.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 82.6%

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

      6. associate-/r/ 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r/ 82.6%

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

   8. Applied egg-rr 82.6%

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

   if -3.9000000000000004e66 < z < 1.45e155

   1. Initial program 92.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in z around 0 89.4%

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

      1. associate-*r/ 89.4%

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

      2. distribute-lft-in 89.4%

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

      3. metadata-eval 89.4%

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

      4. neg-mul-1 89.4%

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

      5. sub-neg 89.4%

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

   6. Simplified 89.4%

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

   if 1.45e155 < z

   1. Initial program 83.8%

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

   2. Taylor expanded in z around inf 72.2%

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

      1. mul-1-neg 72.2%

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

      2. associate-*l/ 79.9%

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

      3. distribute-rgt-neg-out 79.9%

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

   4. Simplified 79.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 33.9%

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

      3. sqr-neg 33.9%

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

      4. sqrt-unprod 79.7%

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

      5. add-sqr-sqrt 79.9%

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

      6. associate-*l/ 72.2%

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

      7. expm1-log1p-u 42.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      8. expm1-udef 30.5%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]

      9. associate-*l/ 37.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{y} \cdot z}\right)} - 1\right| \]

      10. clear-num 37.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot z\right)} - 1\right| \]

      11. associate-*l/ 37.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot z}{\frac{y}{x}}}\right)} - 1\right| \]

      12. *-un-lft-identity 37.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{z}}{\frac{y}{x}}\right)} - 1\right| \]

   6. Applied egg-rr 37.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{z}{\frac{y}{x}}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 41.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{z}{\frac{y}{x}}\right)\right)}\right| \]

      2. expm1-log1p 79.8%

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

      3. associate-/r/ 80.5%

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

      4. *-commutative 80.5%

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

   8. Simplified 80.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+66}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 10: 68.7% 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 88.5%

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

   2. Taylor expanded in z around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. metadata-eval 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 64.7%

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

   if -10.5 < x < 4

   1. Initial program 96.1%

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

   2. Taylor expanded in x around 0 76.3%

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

4. Final simplification 70.7%

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

Alternative 11: 39.8% 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 92.5%

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

2. Taylor expanded in x around 0 42.2%

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

3. Final simplification 42.2%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))