fabs fraction 1

Percentage Accurate: 91.8% → 99.9%

Time: 6.6s

Alternatives: 8

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.008:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 0.008)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ x (/ y z)) (/ (+ x 4.0) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.008) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x / (y / z)) - ((x + 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 (y <= 0.008d0) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x / (y / z)) - ((x + 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 (y <= 0.008) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x / (y / z)) - ((x + 4.0) / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 0.008:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x / (y / z)) - ((x + 4.0) / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 0.008)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x / Float64(y / z)) - Float64(Float64(x + 4.0) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.008)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x / (y / z)) - ((x + 4.0) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 0.008], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0080000000000000002

   1. Initial program 90.6%

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

      1. associate-*l/ 92.7%

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

      2. sub-div 98.0%

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

   3. Applied egg-rr 98.0%

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

   if 0.0080000000000000002 < y

   1. Initial program 94.2%

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

   3. Simplified 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.008:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 2: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -10.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2050:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))) (t_2 (fabs (/ 4.0 y))))
   (if (<= x -3.4e+176)
     t_1
     (if (<= x -2.8e+153)
       t_0
       (if (<= x -10.2)
         t_1
         (if (<= x 1.35e-134)
           t_2
           (if (<= x 5e-114)
             t_0
             (if (<= x 2.5e-82) t_2 (if (<= x 2050.0) 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 t_2 = fabs((4.0 / y));
	double tmp;
	if (x <= -3.4e+176) {
		tmp = t_1;
	} else if (x <= -2.8e+153) {
		tmp = t_0;
	} else if (x <= -10.2) {
		tmp = t_1;
	} else if (x <= 1.35e-134) {
		tmp = t_2;
	} else if (x <= 5e-114) {
		tmp = t_0;
	} else if (x <= 2.5e-82) {
		tmp = t_2;
	} else if (x <= 2050.0) {
		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) :: t_2
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    t_1 = abs((x / y))
    t_2 = abs((4.0d0 / y))
    if (x <= (-3.4d+176)) then
        tmp = t_1
    else if (x <= (-2.8d+153)) then
        tmp = t_0
    else if (x <= (-10.2d0)) then
        tmp = t_1
    else if (x <= 1.35d-134) then
        tmp = t_2
    else if (x <= 5d-114) then
        tmp = t_0
    else if (x <= 2.5d-82) then
        tmp = t_2
    else if (x <= 2050.0d0) 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 t_2 = Math.abs((4.0 / y));
	double tmp;
	if (x <= -3.4e+176) {
		tmp = t_1;
	} else if (x <= -2.8e+153) {
		tmp = t_0;
	} else if (x <= -10.2) {
		tmp = t_1;
	} else if (x <= 1.35e-134) {
		tmp = t_2;
	} else if (x <= 5e-114) {
		tmp = t_0;
	} else if (x <= 2.5e-82) {
		tmp = t_2;
	} else if (x <= 2050.0) {
		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))
	t_2 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -3.4e+176:
		tmp = t_1
	elif x <= -2.8e+153:
		tmp = t_0
	elif x <= -10.2:
		tmp = t_1
	elif x <= 1.35e-134:
		tmp = t_2
	elif x <= 5e-114:
		tmp = t_0
	elif x <= 2.5e-82:
		tmp = t_2
	elif x <= 2050.0:
		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))
	t_2 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -3.4e+176)
		tmp = t_1;
	elseif (x <= -2.8e+153)
		tmp = t_0;
	elseif (x <= -10.2)
		tmp = t_1;
	elseif (x <= 1.35e-134)
		tmp = t_2;
	elseif (x <= 5e-114)
		tmp = t_0;
	elseif (x <= 2.5e-82)
		tmp = t_2;
	elseif (x <= 2050.0)
		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));
	t_2 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -3.4e+176)
		tmp = t_1;
	elseif (x <= -2.8e+153)
		tmp = t_0;
	elseif (x <= -10.2)
		tmp = t_1;
	elseif (x <= 1.35e-134)
		tmp = t_2;
	elseif (x <= 5e-114)
		tmp = t_0;
	elseif (x <= 2.5e-82)
		tmp = t_2;
	elseif (x <= 2050.0)
		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]}, Block[{t$95$2 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e+176], t$95$1, If[LessEqual[x, -2.8e+153], t$95$0, If[LessEqual[x, -10.2], t$95$1, If[LessEqual[x, 1.35e-134], t$95$2, If[LessEqual[x, 5e-114], t$95$0, If[LessEqual[x, 2.5e-82], t$95$2, If[LessEqual[x, 2050.0], t$95$0, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000014e176 or -2.79999999999999985e153 < x < -10.199999999999999 or 2050 < x

   1. Initial program 86.3%

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

   2. Taylor expanded in x around inf 84.4%

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

   3. Taylor expanded in z around 0 75.2%

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

   if -3.40000000000000014e176 < x < -2.79999999999999985e153 or 1.3499999999999999e-134 < x < 4.99999999999999989e-114 or 2.4999999999999999e-82 < x < 2050

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. associate-*r/ 81.6%

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

      3. distribute-rgt-neg-out 81.6%

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

      4. distribute-neg-frac 81.6%

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

   4. Simplified 81.6%

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

      1. associate-*r/ 79.5%

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

      2. distribute-rgt-neg-in 79.5%

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

      3. distribute-neg-frac 79.5%

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

      4. associate-*l/ 83.1%

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

      5. distribute-rgt-neg-in 83.1%

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

      6. add-sqr-sqrt 47.9%

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

      7. sqrt-unprod 46.3%

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

      8. sqr-neg 46.3%

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

      9. sqrt-unprod 34.9%

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

      10. add-sqr-sqrt 83.1%

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

      11. expm1-log1p-u 52.6%

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

      12. expm1-udef 38.4%

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

   6. Applied egg-rr 38.4%

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

      1. expm1-def 52.6%

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

      2. expm1-log1p 81.6%

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

   8. Simplified 81.6%

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

   if -10.199999999999999 < x < 1.3499999999999999e-134 or 4.99999999999999989e-114 < x < 2.4999999999999999e-82

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 82.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+176}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+153}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -10.2:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-82}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2050:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -10.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2050:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))) (t_2 (fabs (/ 4.0 y))))
   (if (<= x -4.7e+176)
     t_1
     (if (<= x -1.8e+156)
       t_0
       (if (<= x -10.5)
         t_1
         (if (<= x 1.35e-134)
           t_2
           (if (<= x 5e-114)
             t_0
             (if (<= x 8e-82)
               t_2
               (if (<= x 2050.0) (fabs (/ x (/ y z))) 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 t_2 = fabs((4.0 / y));
	double tmp;
	if (x <= -4.7e+176) {
		tmp = t_1;
	} else if (x <= -1.8e+156) {
		tmp = t_0;
	} else if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 1.35e-134) {
		tmp = t_2;
	} else if (x <= 5e-114) {
		tmp = t_0;
	} else if (x <= 8e-82) {
		tmp = t_2;
	} else if (x <= 2050.0) {
		tmp = fabs((x / (y / z)));
	} 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) :: t_2
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    t_1 = abs((x / y))
    t_2 = abs((4.0d0 / y))
    if (x <= (-4.7d+176)) then
        tmp = t_1
    else if (x <= (-1.8d+156)) then
        tmp = t_0
    else if (x <= (-10.5d0)) then
        tmp = t_1
    else if (x <= 1.35d-134) then
        tmp = t_2
    else if (x <= 5d-114) then
        tmp = t_0
    else if (x <= 8d-82) then
        tmp = t_2
    else if (x <= 2050.0d0) then
        tmp = abs((x / (y / z)))
    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 t_2 = Math.abs((4.0 / y));
	double tmp;
	if (x <= -4.7e+176) {
		tmp = t_1;
	} else if (x <= -1.8e+156) {
		tmp = t_0;
	} else if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 1.35e-134) {
		tmp = t_2;
	} else if (x <= 5e-114) {
		tmp = t_0;
	} else if (x <= 8e-82) {
		tmp = t_2;
	} else if (x <= 2050.0) {
		tmp = Math.abs((x / (y / z)));
	} 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))
	t_2 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -4.7e+176:
		tmp = t_1
	elif x <= -1.8e+156:
		tmp = t_0
	elif x <= -10.5:
		tmp = t_1
	elif x <= 1.35e-134:
		tmp = t_2
	elif x <= 5e-114:
		tmp = t_0
	elif x <= 8e-82:
		tmp = t_2
	elif x <= 2050.0:
		tmp = math.fabs((x / (y / z)))
	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))
	t_2 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -4.7e+176)
		tmp = t_1;
	elseif (x <= -1.8e+156)
		tmp = t_0;
	elseif (x <= -10.5)
		tmp = t_1;
	elseif (x <= 1.35e-134)
		tmp = t_2;
	elseif (x <= 5e-114)
		tmp = t_0;
	elseif (x <= 8e-82)
		tmp = t_2;
	elseif (x <= 2050.0)
		tmp = abs(Float64(x / Float64(y / z)));
	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));
	t_2 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -4.7e+176)
		tmp = t_1;
	elseif (x <= -1.8e+156)
		tmp = t_0;
	elseif (x <= -10.5)
		tmp = t_1;
	elseif (x <= 1.35e-134)
		tmp = t_2;
	elseif (x <= 5e-114)
		tmp = t_0;
	elseif (x <= 8e-82)
		tmp = t_2;
	elseif (x <= 2050.0)
		tmp = abs((x / (y / z)));
	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]}, Block[{t$95$2 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.7e+176], t$95$1, If[LessEqual[x, -1.8e+156], t$95$0, If[LessEqual[x, -10.5], t$95$1, If[LessEqual[x, 1.35e-134], t$95$2, If[LessEqual[x, 5e-114], t$95$0, If[LessEqual[x, 8e-82], t$95$2, If[LessEqual[x, 2050.0], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.69999999999999981e176 or -1.79999999999999989e156 < x < -10.5 or 2050 < x

   1. Initial program 86.3%

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

   2. Taylor expanded in x around inf 84.4%

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

   3. Taylor expanded in z around 0 75.2%

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

   if -4.69999999999999981e176 < x < -1.79999999999999989e156 or 1.3499999999999999e-134 < x < 4.99999999999999989e-114

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. associate-*r/ 92.7%

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

      3. distribute-rgt-neg-out 92.7%

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

      4. distribute-neg-frac 92.7%

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

   4. Simplified 92.7%

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

      1. associate-*r/ 85.2%

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

      2. distribute-rgt-neg-in 85.2%

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

      3. distribute-neg-frac 85.2%

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

      4. associate-*l/ 92.6%

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

      5. distribute-rgt-neg-in 92.6%

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

      6. add-sqr-sqrt 61.5%

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

      7. sqrt-unprod 48.2%

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

      8. sqr-neg 48.2%

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

      9. sqrt-unprod 30.8%

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

      10. add-sqr-sqrt 92.6%

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

      11. expm1-log1p-u 60.2%

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

      12. expm1-udef 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. expm1-def 60.3%

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

      2. expm1-log1p 92.7%

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

   8. Simplified 92.7%

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

   if -10.5 < x < 1.3499999999999999e-134 or 4.99999999999999989e-114 < x < 8e-82

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 82.5%

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

   if 8e-82 < x < 2050

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. associate-*r/ 70.6%

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

      3. distribute-rgt-neg-out 70.6%

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

      4. distribute-neg-frac 70.6%

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

   4. Simplified 70.6%

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

      1. clear-num 70.6%

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

      2. un-div-inv 73.8%

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

      3. add-sqr-sqrt 34.0%

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

      4. sqrt-unprod 44.4%

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

      5. sqr-neg 44.4%

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

      6. sqrt-unprod 39.1%

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

      7. add-sqr-sqrt 73.8%

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

   6. Applied egg-rr 73.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+176}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+156}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-82}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2050:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -10.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-114}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2050:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y x)))) (t_1 (fabs (/ x y))) (t_2 (fabs (/ 4.0 y))))
   (if (<= x -2.15e+200)
     t_1
     (if (<= x -1.35e+156)
       t_0
       (if (<= x -10.5)
         t_1
         (if (<= x 1.35e-134)
           t_2
           (if (<= x 5.8e-114)
             (fabs (* x (/ z y)))
             (if (<= x 4.4e-82) t_2 (if (<= x 2050.0) t_0 t_1)))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z / (y / x)));
	double t_1 = fabs((x / y));
	double t_2 = fabs((4.0 / y));
	double tmp;
	if (x <= -2.15e+200) {
		tmp = t_1;
	} else if (x <= -1.35e+156) {
		tmp = t_0;
	} else if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 1.35e-134) {
		tmp = t_2;
	} else if (x <= 5.8e-114) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 4.4e-82) {
		tmp = t_2;
	} else if (x <= 2050.0) {
		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) :: t_2
    real(8) :: tmp
    t_0 = abs((z / (y / x)))
    t_1 = abs((x / y))
    t_2 = abs((4.0d0 / y))
    if (x <= (-2.15d+200)) then
        tmp = t_1
    else if (x <= (-1.35d+156)) then
        tmp = t_0
    else if (x <= (-10.5d0)) then
        tmp = t_1
    else if (x <= 1.35d-134) then
        tmp = t_2
    else if (x <= 5.8d-114) then
        tmp = abs((x * (z / y)))
    else if (x <= 4.4d-82) then
        tmp = t_2
    else if (x <= 2050.0d0) 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((z / (y / x)));
	double t_1 = Math.abs((x / y));
	double t_2 = Math.abs((4.0 / y));
	double tmp;
	if (x <= -2.15e+200) {
		tmp = t_1;
	} else if (x <= -1.35e+156) {
		tmp = t_0;
	} else if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 1.35e-134) {
		tmp = t_2;
	} else if (x <= 5.8e-114) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 4.4e-82) {
		tmp = t_2;
	} else if (x <= 2050.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z / (y / x)))
	t_1 = math.fabs((x / y))
	t_2 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -2.15e+200:
		tmp = t_1
	elif x <= -1.35e+156:
		tmp = t_0
	elif x <= -10.5:
		tmp = t_1
	elif x <= 1.35e-134:
		tmp = t_2
	elif x <= 5.8e-114:
		tmp = math.fabs((x * (z / y)))
	elif x <= 4.4e-82:
		tmp = t_2
	elif x <= 2050.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z / Float64(y / x)))
	t_1 = abs(Float64(x / y))
	t_2 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -2.15e+200)
		tmp = t_1;
	elseif (x <= -1.35e+156)
		tmp = t_0;
	elseif (x <= -10.5)
		tmp = t_1;
	elseif (x <= 1.35e-134)
		tmp = t_2;
	elseif (x <= 5.8e-114)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 4.4e-82)
		tmp = t_2;
	elseif (x <= 2050.0)
		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((z / (y / x)));
	t_1 = abs((x / y));
	t_2 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -2.15e+200)
		tmp = t_1;
	elseif (x <= -1.35e+156)
		tmp = t_0;
	elseif (x <= -10.5)
		tmp = t_1;
	elseif (x <= 1.35e-134)
		tmp = t_2;
	elseif (x <= 5.8e-114)
		tmp = abs((x * (z / y)));
	elseif (x <= 4.4e-82)
		tmp = t_2;
	elseif (x <= 2050.0)
		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[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.15e+200], t$95$1, If[LessEqual[x, -1.35e+156], t$95$0, If[LessEqual[x, -10.5], t$95$1, If[LessEqual[x, 1.35e-134], t$95$2, If[LessEqual[x, 5.8e-114], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.4e-82], t$95$2, If[LessEqual[x, 2050.0], t$95$0, t$95$1]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.15000000000000016e200 or -1.35e156 < x < -10.5 or 2050 < x

   1. Initial program 88.1%

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

   2. Taylor expanded in x around inf 86.2%

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

   3. Taylor expanded in z around 0 74.7%

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

   if -2.15000000000000016e200 < x < -1.35e156 or 4.39999999999999971e-82 < x < 2050

   1. Initial program 87.8%

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

   2. Taylor expanded in z around inf 63.9%

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

      1. mul-1-neg 63.9%

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

      2. associate-*r/ 66.1%

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

      3. distribute-rgt-neg-out 66.1%

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

      4. distribute-neg-frac 66.1%

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

   4. Simplified 66.1%

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

      1. associate-*r/ 63.9%

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

      2. distribute-rgt-neg-in 63.9%

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

      3. distribute-neg-frac 63.9%

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

      4. associate-*l/ 86.2%

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

      5. *-commutative 86.2%

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

      6. distribute-lft-neg-in 86.2%

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

      7. add-sqr-sqrt 41.8%

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

      8. sqrt-unprod 51.2%

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

      9. sqr-neg 51.2%

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

      10. sqrt-unprod 44.2%

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

      11. add-sqr-sqrt 86.2%

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

      12. clear-num 86.2%

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

      13. un-div-inv 86.4%

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

   6. Applied egg-rr 86.4%

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

   if -10.5 < x < 1.3499999999999999e-134 or 5.79999999999999993e-114 < x < 4.39999999999999971e-82

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 82.5%

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

   if 1.3499999999999999e-134 < x < 5.79999999999999993e-114

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 86.5%

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

      1. mul-1-neg 86.5%

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

      2. associate-*r/ 86.5%

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

      3. distribute-rgt-neg-out 86.5%

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

      4. distribute-neg-frac 86.5%

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

   4. Simplified 86.5%

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

      1. associate-*r/ 86.5%

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

      2. distribute-rgt-neg-in 86.5%

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

      3. distribute-neg-frac 86.5%

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

      4. associate-*l/ 86.3%

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

      5. distribute-rgt-neg-in 86.3%

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

      6. add-sqr-sqrt 71.6%

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

      7. sqrt-unprod 17.6%

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

      8. sqr-neg 17.6%

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

      9. sqrt-unprod 14.3%

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

      10. add-sqr-sqrt 86.3%

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

      11. expm1-log1p-u 70.4%

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

      12. expm1-udef 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. expm1-def 70.5%

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

      2. expm1-log1p 86.5%

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

   8. Simplified 86.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+200}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{+156}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-114}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-82}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2050:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+18}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+18)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (* (/ x y) (- 1.0 z)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+18) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x / y) * (1.0 - 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 (x <= 1d+18) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x / y) * (1.0d0 - 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 (x <= 1e+18) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= 1e+18:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+18)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1e+18)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x / y) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1e+18], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e18

   1. Initial program 93.5%

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

      1. associate-*l/ 97.3%

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

      2. sub-div 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 1e18 < x

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. *-commutative 85.9%

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

      3. distribute-rgt-out-- 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot \left(1 - z\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+18}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+94}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+94)
   (fabs (* x (/ z y)))
   (if (<= z 1.9e+102) (fabs (/ (- -4.0 x) y)) (fabs (/ x (/ y z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+94) {
		tmp = fabs((x * (z / y)));
	} else if (z <= 1.9e+102) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((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 (z <= (-4.5d+94)) then
        tmp = abs((x * (z / y)))
    else if (z <= 1.9d+102) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((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 (z <= -4.5e+94) {
		tmp = Math.abs((x * (z / y)));
	} else if (z <= 1.9e+102) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -4.5e+94:
		tmp = math.fabs((x * (z / y)))
	elif z <= 1.9e+102:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5e+94)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (z <= 1.9e+102)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.5e+94)
		tmp = abs((x * (z / y)));
	elseif (z <= 1.9e+102)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -4.5e+94], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.9e+102], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.49999999999999972e94

   1. Initial program 94.9%

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

   2. Taylor expanded in z around inf 83.1%

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

      1. mul-1-neg 83.1%

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

      2. associate-*r/ 84.9%

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

      3. distribute-rgt-neg-out 84.9%

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

      4. distribute-neg-frac 84.9%

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

   4. Simplified 84.9%

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

      1. associate-*r/ 83.1%

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

      2. distribute-rgt-neg-in 83.1%

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

      3. distribute-neg-frac 83.1%

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

      4. associate-*l/ 79.8%

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

      5. distribute-rgt-neg-in 79.8%

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

      6. add-sqr-sqrt 79.6%

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

      7. sqrt-unprod 55.7%

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

      8. sqr-neg 55.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 79.8%

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

      11. expm1-log1p-u 42.7%

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

      12. expm1-udef 34.5%

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

   6. Applied egg-rr 34.5%

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

      1. expm1-def 47.8%

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

      2. expm1-log1p 84.9%

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

   8. Simplified 84.9%

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

   if -4.49999999999999972e94 < z < 1.89999999999999989e102

   1. Initial program 93.3%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around 0 94.8%

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

      1. associate-*r/ 94.8%

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

      2. distribute-lft-in 94.8%

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

      3. metadata-eval 94.8%

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

      4. neg-mul-1 94.8%

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

      5. sub-neg 94.8%

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

   6. Simplified 94.8%

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

   if 1.89999999999999989e102 < z

   1. Initial program 80.0%

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

   2. Taylor expanded in z around inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. associate-*r/ 88.3%

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

      3. distribute-rgt-neg-out 88.3%

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

      4. distribute-neg-frac 88.3%

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

   4. Simplified 88.3%

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

      1. clear-num 88.2%

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

      2. un-div-inv 89.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 52.3%

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

      5. sqr-neg 52.3%

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

      6. sqrt-unprod 89.2%

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

      7. add-sqr-sqrt 89.4%

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

   6. Applied egg-rr 89.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+94}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 7: 69.2% 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 86.9%

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

   2. Taylor expanded in x around inf 85.1%

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

   3. Taylor expanded in z around 0 72.0%

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

   if -10.5 < x < 4

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 73.3%

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

4. Final simplification 72.6%

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

Alternative 8: 41.0% 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 91.6%

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

2. Taylor expanded in x around 0 38.7%

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

3. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023299 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))