fabs fraction 1

Percentage Accurate: 91.1% → 99.6%

Time: 6.3s

Alternatives: 11

Speedup: 1.0×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e-70)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-70) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e-70)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1.5e-70], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5000000000000001e-70

   1. Initial program 91.5%

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

      1. associate-*l/ 89.9%

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

      2. sub-div 96.1%

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

   3. Applied egg-rr 96.1%

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

   if 1.5000000000000001e-70 < y

   1. Initial program 98.7%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y) (* z (/ x y)))))
   (if (<= t_0 5e+249) (fabs t_0) (fabs (/ (- (+ x 4.0) (* x z)) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= 5e+249) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y) - (z * (x / y))
    if (t_0 <= 5d+249) then
        tmp = abs(t_0)
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= 5e+249) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = ((x + 4.0) / y) - (z * (x / y))
	tmp = 0
	if t_0 <= 5e+249:
		tmp = math.fabs(t_0)
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y)))
	tmp = 0.0
	if (t_0 <= 5e+249)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = ((x + 4.0) / y) - (z * (x / y));
	tmp = 0.0;
	if (t_0 <= 5e+249)
		tmp = abs(t_0);
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+249], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 4.9999999999999996e249

   1. Initial program 98.6%

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

   if 4.9999999999999996e249 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 67.4%

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

      1. associate-*l/ 70.2%

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

      2. sub-div 97.7%

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

   3. Applied egg-rr 97.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 3: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -110000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y)))) (t_1 (fabs (/ x y))))
   (if (<= x -1.3e+207)
     t_0
     (if (<= x -3.1e+129)
       t_1
       (if (<= x -4.5e+78)
         t_0
         (if (<= x -110000000000.0)
           t_1
           (if (<= x -3.9e-72)
             t_0
             (if (<= x 5.1e-124)
               (fabs (/ 4.0 y))
               (if (<= x 1.22e+141) t_0 t_1)))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x * (z / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.3e+207) {
		tmp = t_0;
	} else if (x <= -3.1e+129) {
		tmp = t_1;
	} else if (x <= -4.5e+78) {
		tmp = t_0;
	} else if (x <= -110000000000.0) {
		tmp = t_1;
	} else if (x <= -3.9e-72) {
		tmp = t_0;
	} else if (x <= 5.1e-124) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.22e+141) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    t_1 = abs((x / y))
    if (x <= (-1.3d+207)) then
        tmp = t_0
    else if (x <= (-3.1d+129)) then
        tmp = t_1
    else if (x <= (-4.5d+78)) then
        tmp = t_0
    else if (x <= (-110000000000.0d0)) then
        tmp = t_1
    else if (x <= (-3.9d-72)) then
        tmp = t_0
    else if (x <= 5.1d-124) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.22d+141) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x * (z / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.3e+207) {
		tmp = t_0;
	} else if (x <= -3.1e+129) {
		tmp = t_1;
	} else if (x <= -4.5e+78) {
		tmp = t_0;
	} else if (x <= -110000000000.0) {
		tmp = t_1;
	} else if (x <= -3.9e-72) {
		tmp = t_0;
	} else if (x <= 5.1e-124) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.22e+141) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x * (z / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.3e+207:
		tmp = t_0
	elif x <= -3.1e+129:
		tmp = t_1
	elif x <= -4.5e+78:
		tmp = t_0
	elif x <= -110000000000.0:
		tmp = t_1
	elif x <= -3.9e-72:
		tmp = t_0
	elif x <= 5.1e-124:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.22e+141:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x * Float64(z / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.3e+207)
		tmp = t_0;
	elseif (x <= -3.1e+129)
		tmp = t_1;
	elseif (x <= -4.5e+78)
		tmp = t_0;
	elseif (x <= -110000000000.0)
		tmp = t_1;
	elseif (x <= -3.9e-72)
		tmp = t_0;
	elseif (x <= 5.1e-124)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.22e+141)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x * (z / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.3e+207)
		tmp = t_0;
	elseif (x <= -3.1e+129)
		tmp = t_1;
	elseif (x <= -4.5e+78)
		tmp = t_0;
	elseif (x <= -110000000000.0)
		tmp = t_1;
	elseif (x <= -3.9e-72)
		tmp = t_0;
	elseif (x <= 5.1e-124)
		tmp = abs((4.0 / y));
	elseif (x <= 1.22e+141)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.3e+207], t$95$0, If[LessEqual[x, -3.1e+129], t$95$1, If[LessEqual[x, -4.5e+78], t$95$0, If[LessEqual[x, -110000000000.0], t$95$1, If[LessEqual[x, -3.9e-72], t$95$0, If[LessEqual[x, 5.1e-124], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.22e+141], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2999999999999999e207 or -3.1e129 < x < -4.4999999999999999e78 or -1.1e11 < x < -3.9e-72 or 5.1000000000000001e-124 < x < 1.2199999999999999e141

   1. Initial program 96.1%

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

   2. Taylor expanded in z around inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. associate-*r/ 67.5%

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

      3. distribute-rgt-neg-out 67.5%

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

      4. distribute-neg-frac 67.5%

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

   4. Simplified 67.5%

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

      1. add-sqr-sqrt 30.7%

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

      2. sqrt-unprod 49.6%

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

      3. sqr-neg 49.6%

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

      4. sqrt-unprod 36.7%

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

      5. add-sqr-sqrt 67.5%

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

      6. clear-num 67.4%

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

      7. expm1-log1p-u 40.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 40.4%

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

      9. expm1-udef 28.6%

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

      10. associate-/r/ 28.6%

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

      11. associate-/r/ 28.6%

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

      12. div-inv 28.6%

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

      13. clear-num 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 40.4%

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

      2. expm1-log1p 67.5%

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

   8. Simplified 67.5%

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

   if -1.2999999999999999e207 < x < -3.1e129 or -4.4999999999999999e78 < x < -1.1e11 or 1.2199999999999999e141 < x

   1. Initial program 83.5%

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

   2. Taylor expanded in z around 0 79.0%

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

      1. associate-*r/ 79.0%

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

      2. metadata-eval 79.0%

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

   4. Simplified 79.0%

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

   5. Taylor expanded in x around inf 79.0%

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

   if -3.9e-72 < x < 5.1000000000000001e-124

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 85.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+207}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+78}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -110000000000:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+141}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -1.4e+207)
     t_0
     (if (<= x -4.4e+129)
       t_1
       (if (<= x -2.8e+72)
         t_0
         (if (<= x -3e+16)
           t_1
           (if (<= x -3.8e-72)
             (fabs (* x (/ z y)))
             (if (<= x 2.55e-124) (fabs (/ 4.0 y)) t_0))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.4e+207) {
		tmp = t_0;
	} else if (x <= -4.4e+129) {
		tmp = t_1;
	} else if (x <= -2.8e+72) {
		tmp = t_0;
	} else if (x <= -3e+16) {
		tmp = t_1;
	} else if (x <= -3.8e-72) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 2.55e-124) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-1.4d+207)) then
        tmp = t_0
    else if (x <= (-4.4d+129)) then
        tmp = t_1
    else if (x <= (-2.8d+72)) then
        tmp = t_0
    else if (x <= (-3d+16)) then
        tmp = t_1
    else if (x <= (-3.8d-72)) then
        tmp = abs((x * (z / y)))
    else if (x <= 2.55d-124) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.4e+207) {
		tmp = t_0;
	} else if (x <= -4.4e+129) {
		tmp = t_1;
	} else if (x <= -2.8e+72) {
		tmp = t_0;
	} else if (x <= -3e+16) {
		tmp = t_1;
	} else if (x <= -3.8e-72) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 2.55e-124) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.4e+207:
		tmp = t_0
	elif x <= -4.4e+129:
		tmp = t_1
	elif x <= -2.8e+72:
		tmp = t_0
	elif x <= -3e+16:
		tmp = t_1
	elif x <= -3.8e-72:
		tmp = math.fabs((x * (z / y)))
	elif x <= 2.55e-124:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.4e+207)
		tmp = t_0;
	elseif (x <= -4.4e+129)
		tmp = t_1;
	elseif (x <= -2.8e+72)
		tmp = t_0;
	elseif (x <= -3e+16)
		tmp = t_1;
	elseif (x <= -3.8e-72)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 2.55e-124)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.4e+207)
		tmp = t_0;
	elseif (x <= -4.4e+129)
		tmp = t_1;
	elseif (x <= -2.8e+72)
		tmp = t_0;
	elseif (x <= -3e+16)
		tmp = t_1;
	elseif (x <= -3.8e-72)
		tmp = abs((x * (z / y)));
	elseif (x <= 2.55e-124)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.4e+207], t$95$0, If[LessEqual[x, -4.4e+129], t$95$1, If[LessEqual[x, -2.8e+72], t$95$0, If[LessEqual[x, -3e+16], t$95$1, If[LessEqual[x, -3.8e-72], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.55e-124], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.40000000000000005e207 or -4.3999999999999999e129 < x < -2.7999999999999999e72 or 2.5500000000000001e-124 < x

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf 61.0%

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

      1. mul-1-neg 61.0%

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

      2. associate-*r/ 66.4%

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

      3. distribute-rgt-neg-out 66.4%

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

      4. distribute-neg-frac 66.4%

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

   4. Simplified 66.4%

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

      1. add-sqr-sqrt 31.9%

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

      2. sqrt-unprod 53.0%

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

      3. sqr-neg 53.0%

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

      4. sqrt-unprod 34.4%

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

      5. add-sqr-sqrt 66.4%

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

      6. clear-num 66.3%

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

      7. expm1-log1p-u 36.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 36.0%

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

      9. expm1-udef 28.3%

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

      10. associate-/r/ 29.1%

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

      11. associate-/r/ 28.3%

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

      12. div-inv 28.3%

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

      13. clear-num 28.3%

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

   6. Applied egg-rr 28.3%

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

      1. expm1-def 36.1%

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

      2. expm1-log1p 66.4%

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

      3. associate-*r/ 61.0%

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

      4. associate-*l/ 67.9%

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

      5. *-commutative 67.9%

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

   8. Simplified 67.9%

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

   if -1.40000000000000005e207 < x < -4.3999999999999999e129 or -2.7999999999999999e72 < x < -3e16

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -3e16 < x < -3.80000000000000002e-72

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. associate-*r/ 68.8%

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

      3. distribute-rgt-neg-out 68.8%

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

      4. distribute-neg-frac 68.8%

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

   4. Simplified 68.8%

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

      1. add-sqr-sqrt 25.8%

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

      2. sqrt-unprod 45.1%

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

      3. sqr-neg 45.1%

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

      4. sqrt-unprod 42.7%

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

      5. add-sqr-sqrt 68.8%

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

      6. clear-num 68.8%

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

      7. expm1-log1p-u 38.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 38.1%

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

      9. expm1-udef 21.9%

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

      10. associate-/r/ 21.9%

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

      11. associate-/r/ 21.9%

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

      12. div-inv 21.9%

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

      13. clear-num 21.9%

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

   6. Applied egg-rr 21.9%

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

      1. expm1-def 38.3%

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

      2. expm1-log1p 68.8%

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

   8. Simplified 68.8%

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

   if -3.80000000000000002e-72 < x < 2.5500000000000001e-124

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 85.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+207}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -2e+207)
     (fabs (/ x (/ y z)))
     (if (<= x -4.9e+129)
       t_1
       (if (<= x -4.8e+72)
         t_0
         (if (<= x -1.06e+15)
           t_1
           (if (<= x -2e-72)
             (fabs (* x (/ z y)))
             (if (<= x 4.6e-124) (fabs (/ 4.0 y)) t_0))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -2e+207) {
		tmp = fabs((x / (y / z)));
	} else if (x <= -4.9e+129) {
		tmp = t_1;
	} else if (x <= -4.8e+72) {
		tmp = t_0;
	} else if (x <= -1.06e+15) {
		tmp = t_1;
	} else if (x <= -2e-72) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 4.6e-124) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-2d+207)) then
        tmp = abs((x / (y / z)))
    else if (x <= (-4.9d+129)) then
        tmp = t_1
    else if (x <= (-4.8d+72)) then
        tmp = t_0
    else if (x <= (-1.06d+15)) then
        tmp = t_1
    else if (x <= (-2d-72)) then
        tmp = abs((x * (z / y)))
    else if (x <= 4.6d-124) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -2e+207) {
		tmp = Math.abs((x / (y / z)));
	} else if (x <= -4.9e+129) {
		tmp = t_1;
	} else if (x <= -4.8e+72) {
		tmp = t_0;
	} else if (x <= -1.06e+15) {
		tmp = t_1;
	} else if (x <= -2e-72) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 4.6e-124) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -2e+207:
		tmp = math.fabs((x / (y / z)))
	elif x <= -4.9e+129:
		tmp = t_1
	elif x <= -4.8e+72:
		tmp = t_0
	elif x <= -1.06e+15:
		tmp = t_1
	elif x <= -2e-72:
		tmp = math.fabs((x * (z / y)))
	elif x <= 4.6e-124:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -2e+207)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (x <= -4.9e+129)
		tmp = t_1;
	elseif (x <= -4.8e+72)
		tmp = t_0;
	elseif (x <= -1.06e+15)
		tmp = t_1;
	elseif (x <= -2e-72)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 4.6e-124)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -2e+207)
		tmp = abs((x / (y / z)));
	elseif (x <= -4.9e+129)
		tmp = t_1;
	elseif (x <= -4.8e+72)
		tmp = t_0;
	elseif (x <= -1.06e+15)
		tmp = t_1;
	elseif (x <= -2e-72)
		tmp = abs((x * (z / y)));
	elseif (x <= 4.6e-124)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e+207], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -4.9e+129], t$95$1, If[LessEqual[x, -4.8e+72], t$95$0, If[LessEqual[x, -1.06e+15], t$95$1, If[LessEqual[x, -2e-72], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.6e-124], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.0000000000000001e207

   1. Initial program 87.8%

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

   2. Taylor expanded in z around inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-*r/ 77.2%

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

      3. distribute-rgt-neg-out 77.2%

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

      4. distribute-neg-frac 77.2%

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

   4. Simplified 77.2%

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

      1. add-sqr-sqrt 36.8%

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

      2. sqrt-unprod 69.7%

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

      3. sqr-neg 69.7%

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

      4. sqrt-unprod 40.4%

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

      5. add-sqr-sqrt 77.2%

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

      6. clear-num 77.2%

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

      7. div-inv 77.4%

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

   6. Applied egg-rr 77.4%

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

   if -2.0000000000000001e207 < x < -4.9e129 or -4.8000000000000002e72 < x < -1.06e15

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -4.9e129 < x < -4.8000000000000002e72 or 4.60000000000000024e-124 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in z around inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. associate-*r/ 63.6%

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

      3. distribute-rgt-neg-out 63.6%

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

      4. distribute-neg-frac 63.6%

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

   4. Simplified 63.6%

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

      1. add-sqr-sqrt 30.7%

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

      2. sqrt-unprod 48.6%

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

      3. sqr-neg 48.6%

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

      4. sqrt-unprod 32.8%

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

      5. add-sqr-sqrt 63.6%

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

      6. clear-num 63.5%

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

      7. expm1-log1p-u 35.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 35.2%

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

      9. expm1-udef 25.5%

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

      10. associate-/r/ 26.5%

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

      11. associate-/r/ 25.5%

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

      12. div-inv 25.5%

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

      13. clear-num 25.5%

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

   6. Applied egg-rr 25.5%

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

      1. expm1-def 35.2%

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

      2. expm1-log1p 63.6%

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

      3. associate-*r/ 60.7%

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

      4. associate-*l/ 65.5%

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

      5. *-commutative 65.5%

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

   8. Simplified 65.5%

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

   if -1.06e15 < x < -1.9999999999999999e-72

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. associate-*r/ 68.8%

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

      3. distribute-rgt-neg-out 68.8%

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

      4. distribute-neg-frac 68.8%

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

   4. Simplified 68.8%

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

      1. add-sqr-sqrt 25.8%

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

      2. sqrt-unprod 45.1%

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

      3. sqr-neg 45.1%

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

      4. sqrt-unprod 42.7%

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

      5. add-sqr-sqrt 68.8%

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

      6. clear-num 68.8%

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

      7. expm1-log1p-u 38.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 38.1%

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

      9. expm1-udef 21.9%

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

      10. associate-/r/ 21.9%

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

      11. associate-/r/ 21.9%

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

      12. div-inv 21.9%

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

      13. clear-num 21.9%

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

   6. Applied egg-rr 21.9%

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

      1. expm1-def 38.3%

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

      2. expm1-log1p 68.8%

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

   8. Simplified 68.8%

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

   if -1.9999999999999999e-72 < x < 4.60000000000000024e-124

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 85.2%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-123}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -1.65e+207)
     (fabs (/ x (/ y z)))
     (if (<= x -1.3e+129)
       t_1
       (if (<= x -6.6e+72)
         t_0
         (if (<= x -1.25e+16)
           t_1
           (if (<= x -6e-123)
             (fabs (/ (* x z) y))
             (if (<= x 5.1e-124) (fabs (/ 4.0 y)) t_0))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.65e+207) {
		tmp = fabs((x / (y / z)));
	} else if (x <= -1.3e+129) {
		tmp = t_1;
	} else if (x <= -6.6e+72) {
		tmp = t_0;
	} else if (x <= -1.25e+16) {
		tmp = t_1;
	} else if (x <= -6e-123) {
		tmp = fabs(((x * z) / y));
	} else if (x <= 5.1e-124) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-1.65d+207)) then
        tmp = abs((x / (y / z)))
    else if (x <= (-1.3d+129)) then
        tmp = t_1
    else if (x <= (-6.6d+72)) then
        tmp = t_0
    else if (x <= (-1.25d+16)) then
        tmp = t_1
    else if (x <= (-6d-123)) then
        tmp = abs(((x * z) / y))
    else if (x <= 5.1d-124) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.65e+207) {
		tmp = Math.abs((x / (y / z)));
	} else if (x <= -1.3e+129) {
		tmp = t_1;
	} else if (x <= -6.6e+72) {
		tmp = t_0;
	} else if (x <= -1.25e+16) {
		tmp = t_1;
	} else if (x <= -6e-123) {
		tmp = Math.abs(((x * z) / y));
	} else if (x <= 5.1e-124) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.65e+207:
		tmp = math.fabs((x / (y / z)))
	elif x <= -1.3e+129:
		tmp = t_1
	elif x <= -6.6e+72:
		tmp = t_0
	elif x <= -1.25e+16:
		tmp = t_1
	elif x <= -6e-123:
		tmp = math.fabs(((x * z) / y))
	elif x <= 5.1e-124:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.65e+207)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (x <= -1.3e+129)
		tmp = t_1;
	elseif (x <= -6.6e+72)
		tmp = t_0;
	elseif (x <= -1.25e+16)
		tmp = t_1;
	elseif (x <= -6e-123)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (x <= 5.1e-124)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.65e+207)
		tmp = abs((x / (y / z)));
	elseif (x <= -1.3e+129)
		tmp = t_1;
	elseif (x <= -6.6e+72)
		tmp = t_0;
	elseif (x <= -1.25e+16)
		tmp = t_1;
	elseif (x <= -6e-123)
		tmp = abs(((x * z) / y));
	elseif (x <= 5.1e-124)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65e+207], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.3e+129], t$95$1, If[LessEqual[x, -6.6e+72], t$95$0, If[LessEqual[x, -1.25e+16], t$95$1, If[LessEqual[x, -6e-123], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.1e-124], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.65e207

   1. Initial program 87.8%

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

   2. Taylor expanded in z around inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-*r/ 77.2%

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

      3. distribute-rgt-neg-out 77.2%

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

      4. distribute-neg-frac 77.2%

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

   4. Simplified 77.2%

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

      1. add-sqr-sqrt 36.8%

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

      2. sqrt-unprod 69.7%

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

      3. sqr-neg 69.7%

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

      4. sqrt-unprod 40.4%

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

      5. add-sqr-sqrt 77.2%

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

      6. clear-num 77.2%

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

      7. div-inv 77.4%

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

   6. Applied egg-rr 77.4%

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

   if -1.65e207 < x < -1.30000000000000006e129 or -6.6e72 < x < -1.25e16

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -1.30000000000000006e129 < x < -6.6e72 or 5.1000000000000001e-124 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in z around inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. associate-*r/ 63.6%

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

      3. distribute-rgt-neg-out 63.6%

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

      4. distribute-neg-frac 63.6%

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

   4. Simplified 63.6%

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

      1. add-sqr-sqrt 30.7%

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

      2. sqrt-unprod 48.6%

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

      3. sqr-neg 48.6%

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

      4. sqrt-unprod 32.8%

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

      5. add-sqr-sqrt 63.6%

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

      6. clear-num 63.5%

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

      7. expm1-log1p-u 35.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 35.2%

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

      9. expm1-udef 25.5%

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

      10. associate-/r/ 26.5%

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

      11. associate-/r/ 25.5%

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

      12. div-inv 25.5%

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

      13. clear-num 25.5%

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

   6. Applied egg-rr 25.5%

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

      1. expm1-def 35.2%

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

      2. expm1-log1p 63.6%

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

      3. associate-*r/ 60.7%

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

      4. associate-*l/ 65.5%

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

      5. *-commutative 65.5%

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

   8. Simplified 65.5%

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

   if -1.25e16 < x < -5.99999999999999968e-123

   1. Initial program 96.8%

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

   2. Taylor expanded in z around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. associate-*r/ 61.7%

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

      3. distribute-rgt-neg-out 61.7%

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

      4. distribute-neg-frac 61.7%

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

   4. Simplified 61.7%

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

      1. associate-*r/ 64.5%

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

      2. add-sqr-sqrt 20.3%

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

      3. times-frac 20.3%

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

      4. add-sqr-sqrt 7.1%

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

      5. sqrt-unprod 8.3%

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

      6. sqr-neg 8.3%

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

      7. sqrt-unprod 13.2%

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

      8. add-sqr-sqrt 20.3%

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

      9. times-frac 20.3%

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

      10. add-sqr-sqrt 64.5%

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

   6. Applied egg-rr 64.5%

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

   if -5.99999999999999968e-123 < x < 5.1000000000000001e-124

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 90.3%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-123}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.65e-45)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-45) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.65d-45) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-45) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2.65e-45:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.65e-45)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.65e-45)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2.65e-45], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6499999999999999e-45

   1. Initial program 91.9%

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

      1. associate-*l/ 90.4%

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

      2. sub-div 96.3%

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

   3. Applied egg-rr 96.3%

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

   if 2.6499999999999999e-45 < y

   1. Initial program 98.6%

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

   3. Simplified 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 8: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+40}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 37000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+40)
   (fabs (* z (/ x y)))
   (if (<= z 37000000000.0) (fabs (/ (- -4.0 x) y)) (fabs (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+40) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 37000000000.0) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((x * (z / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.5d+40)) then
        tmp = abs((z * (x / y)))
    else if (z <= 37000000000.0d0) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+40) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 37000000000.0) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -4.5e+40:
		tmp = math.fabs((z * (x / y)))
	elif z <= 37000000000.0:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5e+40)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 37000000000.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.5e+40)
		tmp = abs((z * (x / y)));
	elseif (z <= 37000000000.0)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -4.5e+40], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 37000000000.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.50000000000000032e40

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. associate-*r/ 73.3%

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

      3. distribute-rgt-neg-out 73.3%

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

      4. distribute-neg-frac 73.3%

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

   4. Simplified 73.3%

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

      1. add-sqr-sqrt 73.1%

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

      2. sqrt-unprod 56.2%

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

      3. sqr-neg 56.2%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 73.3%

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

      6. clear-num 73.2%

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

      7. expm1-log1p-u 42.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 42.6%

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

      9. expm1-udef 31.1%

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

      10. associate-/r/ 34.1%

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

      11. associate-/r/ 31.1%

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

      12. div-inv 31.1%

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

      13. clear-num 31.1%

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

   6. Applied egg-rr 31.1%

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

      1. expm1-def 42.7%

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

      2. expm1-log1p 73.3%

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

      3. associate-*r/ 69.8%

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

      4. associate-*l/ 78.2%

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

      5. *-commutative 78.2%

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

   8. Simplified 78.2%

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

   if -4.50000000000000032e40 < z < 3.7e10

   1. Initial program 96.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 96.8%

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

      1. associate-*r/ 96.8%

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

      2. distribute-lft-in 96.8%

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

      3. metadata-eval 96.8%

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

      4. neg-mul-1 96.8%

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

      5. sub-neg 96.8%

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

   6. Simplified 96.8%

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

   if 3.7e10 < z

   1. Initial program 84.7%

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

   2. Taylor expanded in z around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. associate-*r/ 83.2%

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

      3. distribute-rgt-neg-out 83.2%

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

      4. distribute-neg-frac 83.2%

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

   4. Simplified 83.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 57.9%

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

      3. sqr-neg 57.9%

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

      4. sqrt-unprod 83.0%

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

      5. add-sqr-sqrt 83.2%

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

      6. clear-num 83.2%

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

      7. expm1-log1p-u 45.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 45.7%

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

      9. expm1-udef 33.2%

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

      10. associate-/r/ 33.2%

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

      11. associate-/r/ 33.2%

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

      12. div-inv 33.1%

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

      13. clear-num 33.1%

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

   6. Applied egg-rr 33.1%

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

      1. expm1-def 45.6%

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

      2. expm1-log1p 83.2%

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

   8. Simplified 83.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+40}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 37000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 9: 96.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (fabs (/ (- (+ x 4.0) (* x z)) y)))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) - (x * z)) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 93.7%

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

   1. associate-*l/ 92.3%

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

   2. sub-div 96.5%

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

3. Applied egg-rr 96.5%

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

4. Final simplification 96.5%

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

Alternative 10: 69.4% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in z around 0 58.8%

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

      1. associate-*r/ 58.8%

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

      2. metadata-eval 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in x around inf 58.2%

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

   if -10.5 < x < 4

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0 70.3%

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

4. Final simplification 64.3%

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

Alternative 11: 40.6% accurate, 1.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in x around 0 38.3%

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

3. Final simplification 38.3%

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

Reproduce

herbie shell --seed 2023332 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))