fabs fraction 1

Percentage Accurate: 91.4% → 99.8%

Time: 10.0s

Alternatives: 14

Speedup: 1.0×

• 20.6% 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: 91.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y_m} - \frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-29)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (- (/ (+ 4.0 x) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-29) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 5d-29) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((((4.0d0 + x) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-29) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 5e-29:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-29)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 5e-29)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((((4.0 + x) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5e-29], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999986e-29

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 0 96.4%

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

   if 4.99999999999999986e-29 < y

   1. Initial program 96.8%

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

      1. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+100} \lor \neg \left(x \leq 8.2 \cdot 10^{+236}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -1.95e+139)
     t_0
     (if (<= x -0.047)
       t_1
       (if (<= x -2e-44)
         t_0
         (if (<= x 7.5e-50)
           (fabs (/ 4.0 y_m))
           (if (or (<= x 4e+100) (not (<= x 8.2e+236))) t_0 t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -1.95e+139) {
		tmp = t_0;
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -2e-44) {
		tmp = t_0;
	} else if (x <= 7.5e-50) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 4e+100) || !(x <= 8.2e+236)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-1.95d+139)) then
        tmp = t_0
    else if (x <= (-0.047d0)) then
        tmp = t_1
    else if (x <= (-2d-44)) then
        tmp = t_0
    else if (x <= 7.5d-50) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 4d+100) .or. (.not. (x <= 8.2d+236))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.95e+139) {
		tmp = t_0;
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -2e-44) {
		tmp = t_0;
	} else if (x <= 7.5e-50) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 4e+100) || !(x <= 8.2e+236)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.95e+139:
		tmp = t_0
	elif x <= -0.047:
		tmp = t_1
	elif x <= -2e-44:
		tmp = t_0
	elif x <= 7.5e-50:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 4e+100) or not (x <= 8.2e+236):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.95e+139)
		tmp = t_0;
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -2e-44)
		tmp = t_0;
	elseif (x <= 7.5e-50)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 4e+100) || !(x <= 8.2e+236))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.95e+139)
		tmp = t_0;
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -2e-44)
		tmp = t_0;
	elseif (x <= 7.5e-50)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 4e+100) || ~((x <= 8.2e+236)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.95e+139], t$95$0, If[LessEqual[x, -0.047], t$95$1, If[LessEqual[x, -2e-44], t$95$0, If[LessEqual[x, 7.5e-50], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 4e+100], N[Not[LessEqual[x, 8.2e+236]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95000000000000003e139 or -0.047 < x < -1.99999999999999991e-44 or 7.5e-50 < x < 4.00000000000000006e100 or 8.2000000000000008e236 < x

   1. Initial program 85.8%

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

   3. Taylor expanded in z around inf 57.2%

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

      1. mul-1-neg 57.2%

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

      2. associate-*l/ 73.3%

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

      3. distribute-rgt-neg-out 73.3%

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

   5. Simplified 73.3%

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

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 53.9%

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

      3. sqr-neg 53.9%

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

      4. sqrt-unprod 33.4%

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

      5. add-sqr-sqrt 73.3%

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

      6. expm1-log1p-u 34.4%

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

      7. expm1-udef 28.1%

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

      8. associate-*l/ 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. expm1-def 31.3%

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

      2. expm1-log1p 57.2%

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

      3. associate-*l/ 73.3%

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

   9. Simplified 73.3%

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

   if -1.95000000000000003e139 < x < -0.047 or 4.00000000000000006e100 < x < 8.2000000000000008e236

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. metadata-eval 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around inf 67.0%

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

   if -1.99999999999999991e-44 < x < 7.5e-50

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+139}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-44}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+100} \lor \neg \left(x \leq 8.2 \cdot 10^{+236}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+100}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+238}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -3.1e+132)
     t_0
     (if (<= x -0.047)
       t_1
       (if (<= x -1.1e-43)
         t_0
         (if (<= x 2.9e-60)
           (fabs (/ 4.0 y_m))
           (if (<= x 8.5e+100)
             (fabs (/ x (/ y_m z)))
             (if (<= x 1.45e+238) t_1 t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -3.1e+132) {
		tmp = t_0;
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -1.1e-43) {
		tmp = t_0;
	} else if (x <= 2.9e-60) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 8.5e+100) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= 1.45e+238) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-3.1d+132)) then
        tmp = t_0
    else if (x <= (-0.047d0)) then
        tmp = t_1
    else if (x <= (-1.1d-43)) then
        tmp = t_0
    else if (x <= 2.9d-60) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 8.5d+100) then
        tmp = abs((x / (y_m / z)))
    else if (x <= 1.45d+238) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -3.1e+132) {
		tmp = t_0;
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -1.1e-43) {
		tmp = t_0;
	} else if (x <= 2.9e-60) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 8.5e+100) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= 1.45e+238) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -3.1e+132:
		tmp = t_0
	elif x <= -0.047:
		tmp = t_1
	elif x <= -1.1e-43:
		tmp = t_0
	elif x <= 2.9e-60:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 8.5e+100:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= 1.45e+238:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -3.1e+132)
		tmp = t_0;
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -1.1e-43)
		tmp = t_0;
	elseif (x <= 2.9e-60)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 8.5e+100)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= 1.45e+238)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -3.1e+132)
		tmp = t_0;
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -1.1e-43)
		tmp = t_0;
	elseif (x <= 2.9e-60)
		tmp = abs((4.0 / y_m));
	elseif (x <= 8.5e+100)
		tmp = abs((x / (y_m / z)));
	elseif (x <= 1.45e+238)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.1e+132], t$95$0, If[LessEqual[x, -0.047], t$95$1, If[LessEqual[x, -1.1e-43], t$95$0, If[LessEqual[x, 2.9e-60], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 8.5e+100], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.45e+238], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.0999999999999998e132 or -0.047 < x < -1.09999999999999999e-43 or 1.4500000000000001e238 < x

   1. Initial program 81.0%

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

   3. Taylor expanded in z around inf 50.8%

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

      1. mul-1-neg 50.8%

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

      2. associate-*l/ 77.0%

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

      3. distribute-rgt-neg-out 77.0%

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

   5. Simplified 77.0%

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

      1. add-sqr-sqrt 38.4%

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

      2. sqrt-unprod 56.7%

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

      3. sqr-neg 56.7%

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

      4. sqrt-unprod 38.4%

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

      5. add-sqr-sqrt 77.0%

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

      6. expm1-log1p-u 32.7%

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

      7. expm1-udef 26.0%

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

      8. associate-*l/ 19.4%

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

   7. Applied egg-rr 19.4%

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

      1. expm1-def 26.1%

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

      2. expm1-log1p 50.8%

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

      3. associate-*l/ 77.0%

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

   9. Simplified 77.0%

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

   if -3.0999999999999998e132 < x < -0.047 or 8.50000000000000043e100 < x < 1.4500000000000001e238

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. metadata-eval 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around inf 67.0%

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

   if -1.09999999999999999e-43 < x < 2.8999999999999999e-60

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 2.8999999999999999e-60 < x < 8.50000000000000043e100

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf 70.2%

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

      1. mul-1-neg 70.2%

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

      2. associate-*l/ 65.9%

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

      3. distribute-rgt-neg-out 65.9%

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

   5. Simplified 65.9%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 70.0%

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

      3. add-sqr-sqrt 42.2%

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

      4. sqrt-unprod 48.2%

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

      5. sqr-neg 48.2%

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

      6. sqrt-unprod 27.6%

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

      7. add-sqr-sqrt 70.0%

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

   7. Applied egg-rr 70.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+132}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+100}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+238}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+132}:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+101}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -6.6e+132)
     (fabs (/ z (/ y_m x)))
     (if (<= x -0.047)
       t_1
       (if (<= x -1.45e-43)
         t_0
         (if (<= x 1.5e-53)
           (fabs (/ 4.0 y_m))
           (if (<= x 4.5e+101)
             (fabs (/ x (/ y_m z)))
             (if (<= x 1.9e+234) t_1 t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -6.6e+132) {
		tmp = fabs((z / (y_m / x)));
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -1.45e-43) {
		tmp = t_0;
	} else if (x <= 1.5e-53) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 4.5e+101) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= 1.9e+234) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-6.6d+132)) then
        tmp = abs((z / (y_m / x)))
    else if (x <= (-0.047d0)) then
        tmp = t_1
    else if (x <= (-1.45d-43)) then
        tmp = t_0
    else if (x <= 1.5d-53) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 4.5d+101) then
        tmp = abs((x / (y_m / z)))
    else if (x <= 1.9d+234) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -6.6e+132) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -1.45e-43) {
		tmp = t_0;
	} else if (x <= 1.5e-53) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 4.5e+101) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= 1.9e+234) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -6.6e+132:
		tmp = math.fabs((z / (y_m / x)))
	elif x <= -0.047:
		tmp = t_1
	elif x <= -1.45e-43:
		tmp = t_0
	elif x <= 1.5e-53:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 4.5e+101:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= 1.9e+234:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -6.6e+132)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -1.45e-43)
		tmp = t_0;
	elseif (x <= 1.5e-53)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 4.5e+101)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= 1.9e+234)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -6.6e+132)
		tmp = abs((z / (y_m / x)));
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -1.45e-43)
		tmp = t_0;
	elseif (x <= 1.5e-53)
		tmp = abs((4.0 / y_m));
	elseif (x <= 4.5e+101)
		tmp = abs((x / (y_m / z)));
	elseif (x <= 1.9e+234)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.6e+132], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -0.047], t$95$1, If[LessEqual[x, -1.45e-43], t$95$0, If[LessEqual[x, 1.5e-53], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.5e+101], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.9e+234], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.6000000000000006e132

   1. Initial program 76.3%

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

   3. Taylor expanded in z around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. associate-*l/ 80.2%

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

      3. distribute-rgt-neg-out 80.2%

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

   5. Simplified 80.2%

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

      1. add-sqr-sqrt 41.7%

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

      2. clear-num 41.7%

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

      3. sqrt-unprod 62.9%

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

      4. sqr-neg 62.9%

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

      5. sqrt-unprod 38.4%

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

      6. add-sqr-sqrt 80.2%

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

      7. associate-*l/ 80.3%

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

      8. *-un-lft-identity 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -6.6000000000000006e132 < x < -0.047 or 4.5000000000000002e101 < x < 1.9e234

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. metadata-eval 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around inf 67.0%

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

   if -0.047 < x < -1.4500000000000001e-43 or 1.9e234 < x

   1. Initial program 89.4%

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

   3. Taylor expanded in z around inf 52.0%

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

      1. mul-1-neg 52.0%

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

      2. associate-*l/ 71.1%

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

      3. distribute-rgt-neg-out 71.1%

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

   5. Simplified 71.1%

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

      1. add-sqr-sqrt 32.5%

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

      2. sqrt-unprod 45.6%

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

      3. sqr-neg 45.6%

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

      4. sqrt-unprod 38.4%

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

      5. add-sqr-sqrt 71.1%

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

      6. expm1-log1p-u 32.9%

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

      7. expm1-udef 24.9%

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

      8. associate-*l/ 15.8%

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

   7. Applied egg-rr 15.8%

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

      1. expm1-def 23.6%

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

      2. expm1-log1p 52.0%

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

      3. associate-*l/ 71.1%

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

   9. Simplified 71.1%

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

   if -1.4500000000000001e-43 < x < 1.5000000000000001e-53

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 1.5000000000000001e-53 < x < 4.5000000000000002e101

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf 70.2%

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

      1. mul-1-neg 70.2%

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

      2. associate-*l/ 65.9%

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

      3. distribute-rgt-neg-out 65.9%

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

   5. Simplified 65.9%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 70.0%

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

      3. add-sqr-sqrt 42.2%

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

      4. sqrt-unprod 48.2%

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

      5. sqr-neg 48.2%

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

      6. sqrt-unprod 27.6%

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

      7. add-sqr-sqrt 70.0%

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

   7. Applied egg-rr 70.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+132}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+101}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+234}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+101}:\\ \;\;\;\;\left|\frac{x \cdot z}{y_m}\right|\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -2.9e+140)
     (fabs (/ z (/ y_m x)))
     (if (<= x -0.047)
       t_1
       (if (<= x -1.25e-43)
         t_0
         (if (<= x 7e-50)
           (fabs (/ 4.0 y_m))
           (if (<= x 5.6e+101)
             (fabs (/ (* x z) y_m))
             (if (<= x 4.2e+234) t_1 t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -2.9e+140) {
		tmp = fabs((z / (y_m / x)));
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -1.25e-43) {
		tmp = t_0;
	} else if (x <= 7e-50) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 5.6e+101) {
		tmp = fabs(((x * z) / y_m));
	} else if (x <= 4.2e+234) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-2.9d+140)) then
        tmp = abs((z / (y_m / x)))
    else if (x <= (-0.047d0)) then
        tmp = t_1
    else if (x <= (-1.25d-43)) then
        tmp = t_0
    else if (x <= 7d-50) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 5.6d+101) then
        tmp = abs(((x * z) / y_m))
    else if (x <= 4.2d+234) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -2.9e+140) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (x <= -0.047) {
		tmp = t_1;
	} else if (x <= -1.25e-43) {
		tmp = t_0;
	} else if (x <= 7e-50) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 5.6e+101) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (x <= 4.2e+234) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -2.9e+140:
		tmp = math.fabs((z / (y_m / x)))
	elif x <= -0.047:
		tmp = t_1
	elif x <= -1.25e-43:
		tmp = t_0
	elif x <= 7e-50:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 5.6e+101:
		tmp = math.fabs(((x * z) / y_m))
	elif x <= 4.2e+234:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -2.9e+140)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -1.25e-43)
		tmp = t_0;
	elseif (x <= 7e-50)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 5.6e+101)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (x <= 4.2e+234)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -2.9e+140)
		tmp = abs((z / (y_m / x)));
	elseif (x <= -0.047)
		tmp = t_1;
	elseif (x <= -1.25e-43)
		tmp = t_0;
	elseif (x <= 7e-50)
		tmp = abs((4.0 / y_m));
	elseif (x <= 5.6e+101)
		tmp = abs(((x * z) / y_m));
	elseif (x <= 4.2e+234)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.9e+140], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -0.047], t$95$1, If[LessEqual[x, -1.25e-43], t$95$0, If[LessEqual[x, 7e-50], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.6e+101], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.2e+234], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.8999999999999999e140

   1. Initial program 76.3%

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

   3. Taylor expanded in z around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. associate-*l/ 80.2%

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

      3. distribute-rgt-neg-out 80.2%

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

   5. Simplified 80.2%

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

      1. add-sqr-sqrt 41.7%

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

      2. clear-num 41.7%

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

      3. sqrt-unprod 62.9%

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

      4. sqr-neg 62.9%

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

      5. sqrt-unprod 38.4%

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

      6. add-sqr-sqrt 80.2%

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

      7. associate-*l/ 80.3%

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

      8. *-un-lft-identity 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -2.8999999999999999e140 < x < -0.047 or 5.59999999999999962e101 < x < 4.2e234

   1. Initial program 91.6%

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

   3. Taylor expanded in z around 0 71.5%

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

      1. associate-*r/ 71.5%

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

      2. metadata-eval 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around inf 67.0%

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

   if -0.047 < x < -1.25000000000000005e-43 or 4.2e234 < x

   1. Initial program 89.4%

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

   3. Taylor expanded in z around inf 52.0%

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

      1. mul-1-neg 52.0%

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

      2. associate-*l/ 71.1%

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

      3. distribute-rgt-neg-out 71.1%

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

   5. Simplified 71.1%

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

      1. add-sqr-sqrt 32.5%

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

      2. sqrt-unprod 45.6%

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

      3. sqr-neg 45.6%

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

      4. sqrt-unprod 38.4%

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

      5. add-sqr-sqrt 71.1%

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

      6. expm1-log1p-u 32.9%

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

      7. expm1-udef 24.9%

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

      8. associate-*l/ 15.8%

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

   7. Applied egg-rr 15.8%

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

      1. expm1-def 23.6%

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

      2. expm1-log1p 52.0%

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

      3. associate-*l/ 71.1%

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

   9. Simplified 71.1%

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

   if -1.25000000000000005e-43 < x < 6.99999999999999993e-50

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 6.99999999999999993e-50 < x < 5.59999999999999962e101

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf 70.2%

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

      1. mul-1-neg 70.2%

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

      2. associate-*l/ 65.9%

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

      3. distribute-rgt-neg-out 65.9%

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

   5. Simplified 65.9%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-unprod 48.2%

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

      3. sqr-neg 48.2%

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

      4. sqrt-unprod 23.2%

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

      5. add-sqr-sqrt 65.9%

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

      6. associate-*l/ 70.2%

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

   7. Applied egg-rr 70.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -0.047:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+101}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+234}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|x \cdot \frac{1 - z}{y_m}\right|\\ \mathbf{if}\;x \leq -8.3 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ (- 1.0 z) y_m)))))
   (if (<= x -8.3e+15)
     t_0
     (if (<= x -9.6e-31)
       (fabs (/ (- -4.0 x) y_m))
       (if (<= x -1.3e-43)
         (fabs (* z (/ x y_m)))
         (if (<= x 4e-54) (fabs (/ 4.0 y_m)) t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x * ((1.0 - z) / y_m)));
	double tmp;
	if (x <= -8.3e+15) {
		tmp = t_0;
	} else if (x <= -9.6e-31) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else if (x <= -1.3e-43) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= 4e-54) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x * ((1.0d0 - z) / y_m)))
    if (x <= (-8.3d+15)) then
        tmp = t_0
    else if (x <= (-9.6d-31)) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else if (x <= (-1.3d-43)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= 4d-54) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x * ((1.0 - z) / y_m)));
	double tmp;
	if (x <= -8.3e+15) {
		tmp = t_0;
	} else if (x <= -9.6e-31) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else if (x <= -1.3e-43) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= 4e-54) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x * ((1.0 - z) / y_m)))
	tmp = 0
	if x <= -8.3e+15:
		tmp = t_0
	elif x <= -9.6e-31:
		tmp = math.fabs(((-4.0 - x) / y_m))
	elif x <= -1.3e-43:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= 4e-54:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)))
	tmp = 0.0
	if (x <= -8.3e+15)
		tmp = t_0;
	elseif (x <= -9.6e-31)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	elseif (x <= -1.3e-43)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= 4e-54)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x * ((1.0 - z) / y_m)));
	tmp = 0.0;
	if (x <= -8.3e+15)
		tmp = t_0;
	elseif (x <= -9.6e-31)
		tmp = abs(((-4.0 - x) / y_m));
	elseif (x <= -1.3e-43)
		tmp = abs((z * (x / y_m)));
	elseif (x <= 4e-54)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8.3e+15], t$95$0, If[LessEqual[x, -9.6e-31], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.3e-43], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4e-54], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.3e15 or 4.0000000000000001e-54 < x

   1. Initial program 86.4%

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

      1. associate-/r/ 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. clear-num 92.7%

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

      2. associate-/r/ 91.9%

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

      3. clear-num 91.9%

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

   6. Applied egg-rr 91.9%

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

   7. Taylor expanded in x around inf 96.4%

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

      1. div-sub 96.4%

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

   9. Simplified 96.4%

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

   if -8.3e15 < x < -9.6000000000000001e-31

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 89.9%

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

      1. associate-*r/ 89.9%

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

      2. distribute-lft-in 89.9%

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

      3. metadata-eval 89.9%

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

      4. neg-mul-1 89.9%

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

      5. sub-neg 89.9%

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

   7. Simplified 89.9%

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

   if -9.6000000000000001e-31 < x < -1.3e-43

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*l/ 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

   5. Simplified 100.0%

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

      1. add-sqr-sqrt 79.4%

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

      2. sqrt-unprod 61.1%

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

      3. sqr-neg 61.1%

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

      4. sqrt-unprod 20.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. expm1-log1p-u 77.8%

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

      7. expm1-udef 51.6%

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

      8. associate-*l/ 51.6%

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

   7. Applied egg-rr 51.6%

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

      1. expm1-def 77.5%

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

      2. expm1-log1p 99.7%

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

      3. associate-*l/ 100.0%

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

   9. Simplified 100.0%

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

   if -1.3e-43 < x < 4.0000000000000001e-54

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{+15}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y_m} \cdot \left(z + 1\right)\right|\\ \mathbf{elif}\;x \leq -7.7 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-44}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.3e+15)
   (fabs (* (/ x y_m) (+ z 1.0)))
   (if (<= x -7.7e-31)
     (fabs (/ (- -4.0 x) y_m))
     (if (<= x -4.1e-44)
       (fabs (* z (/ x y_m)))
       (if (<= x 4.1e-50)
         (fabs (/ 4.0 y_m))
         (fabs (* x (/ (- 1.0 z) y_m))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.3e+15) {
		tmp = fabs(((x / y_m) * (z + 1.0)));
	} else if (x <= -7.7e-31) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else if (x <= -4.1e-44) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= 4.1e-50) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.3d+15)) then
        tmp = abs(((x / y_m) * (z + 1.0d0)))
    else if (x <= (-7.7d-31)) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else if (x <= (-4.1d-44)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= 4.1d-50) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.3e+15) {
		tmp = Math.abs(((x / y_m) * (z + 1.0)));
	} else if (x <= -7.7e-31) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else if (x <= -4.1e-44) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= 4.1e-50) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -8.3e+15:
		tmp = math.fabs(((x / y_m) * (z + 1.0)))
	elif x <= -7.7e-31:
		tmp = math.fabs(((-4.0 - x) / y_m))
	elif x <= -4.1e-44:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= 4.1e-50:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -8.3e+15)
		tmp = abs(Float64(Float64(x / y_m) * Float64(z + 1.0)));
	elseif (x <= -7.7e-31)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	elseif (x <= -4.1e-44)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= 4.1e-50)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -8.3e+15)
		tmp = abs(((x / y_m) * (z + 1.0)));
	elseif (x <= -7.7e-31)
		tmp = abs(((-4.0 - x) / y_m));
	elseif (x <= -4.1e-44)
		tmp = abs((z * (x / y_m)));
	elseif (x <= 4.1e-50)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -8.3e+15], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -7.7e-31], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -4.1e-44], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.1e-50], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.3e15

   1. Initial program 82.1%

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

   3. Taylor expanded in x around inf 82.1%

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

      1. sub-neg 82.1%

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

      2. distribute-rgt-neg-out 82.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. sqrt-unprod 86.5%

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

      5. sqr-neg 86.5%

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

      6. sqrt-unprod 46.9%

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

      7. add-sqr-sqrt 91.9%

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

      8. associate-*l/ 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. *-rgt-identity 83.1%

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

      2. associate-*l/ 91.9%

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

      3. distribute-lft-in 99.8%

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

   7. Simplified 99.8%

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

   if -8.3e15 < x < -7.70000000000000012e-31

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 89.9%

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

      1. associate-*r/ 89.9%

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

      2. distribute-lft-in 89.9%

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

      3. metadata-eval 89.9%

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

      4. neg-mul-1 89.9%

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

      5. sub-neg 89.9%

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

   7. Simplified 89.9%

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

   if -7.70000000000000012e-31 < x < -4.09999999999999992e-44

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*l/ 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

   5. Simplified 100.0%

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

      1. add-sqr-sqrt 79.4%

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

      2. sqrt-unprod 61.1%

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

      3. sqr-neg 61.1%

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

      4. sqrt-unprod 20.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. expm1-log1p-u 77.8%

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

      7. expm1-udef 51.6%

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

      8. associate-*l/ 51.6%

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

   7. Applied egg-rr 51.6%

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

      1. expm1-def 77.5%

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

      2. expm1-log1p 99.7%

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

      3. associate-*l/ 100.0%

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

   9. Simplified 100.0%

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

   if -4.09999999999999992e-44 < x < 4.09999999999999985e-50

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 4.09999999999999985e-50 < x

   1. Initial program 90.0%

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

      1. associate-/r/ 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. clear-num 94.8%

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

      2. associate-/r/ 93.5%

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

      3. clear-num 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in x around inf 93.7%

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

      1. div-sub 93.7%

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

   9. Simplified 93.7%

      \[\leadsto \left|\color{blue}{x \cdot \frac{1 - z}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + 1\right)\right|\\ \mathbf{elif}\;x \leq -7.7 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-44}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y_m} \cdot \left(z + 1\right)\right|\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{x}{y_m} + \frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-45}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.3e+15)
   (fabs (* (/ x y_m) (+ z 1.0)))
   (if (<= x -8.2e-31)
     (fabs (+ (/ x y_m) (/ 4.0 y_m)))
     (if (<= x -5.4e-45)
       (fabs (* z (/ x y_m)))
       (if (<= x 8.5e-52)
         (fabs (/ 4.0 y_m))
         (fabs (* x (/ (- 1.0 z) y_m))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.3e+15) {
		tmp = fabs(((x / y_m) * (z + 1.0)));
	} else if (x <= -8.2e-31) {
		tmp = fabs(((x / y_m) + (4.0 / y_m)));
	} else if (x <= -5.4e-45) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= 8.5e-52) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.3d+15)) then
        tmp = abs(((x / y_m) * (z + 1.0d0)))
    else if (x <= (-8.2d-31)) then
        tmp = abs(((x / y_m) + (4.0d0 / y_m)))
    else if (x <= (-5.4d-45)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= 8.5d-52) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.3e+15) {
		tmp = Math.abs(((x / y_m) * (z + 1.0)));
	} else if (x <= -8.2e-31) {
		tmp = Math.abs(((x / y_m) + (4.0 / y_m)));
	} else if (x <= -5.4e-45) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= 8.5e-52) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -8.3e+15:
		tmp = math.fabs(((x / y_m) * (z + 1.0)))
	elif x <= -8.2e-31:
		tmp = math.fabs(((x / y_m) + (4.0 / y_m)))
	elif x <= -5.4e-45:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= 8.5e-52:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -8.3e+15)
		tmp = abs(Float64(Float64(x / y_m) * Float64(z + 1.0)));
	elseif (x <= -8.2e-31)
		tmp = abs(Float64(Float64(x / y_m) + Float64(4.0 / y_m)));
	elseif (x <= -5.4e-45)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= 8.5e-52)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -8.3e+15)
		tmp = abs(((x / y_m) * (z + 1.0)));
	elseif (x <= -8.2e-31)
		tmp = abs(((x / y_m) + (4.0 / y_m)));
	elseif (x <= -5.4e-45)
		tmp = abs((z * (x / y_m)));
	elseif (x <= 8.5e-52)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -8.3e+15], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -8.2e-31], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] + N[(4.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -5.4e-45], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 8.5e-52], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.3e15

   1. Initial program 82.1%

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

   3. Taylor expanded in x around inf 82.1%

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

      1. sub-neg 82.1%

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

      2. distribute-rgt-neg-out 82.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. sqrt-unprod 86.5%

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

      5. sqr-neg 86.5%

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

      6. sqrt-unprod 46.9%

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

      7. add-sqr-sqrt 91.9%

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

      8. associate-*l/ 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. *-rgt-identity 83.1%

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

      2. associate-*l/ 91.9%

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

      3. distribute-lft-in 99.8%

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

   7. Simplified 99.8%

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

   if -8.3e15 < x < -8.1999999999999993e-31

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 90.0%

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

      1. associate-*r/ 90.0%

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

      2. metadata-eval 90.0%

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

   5. Simplified 90.0%

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

   if -8.1999999999999993e-31 < x < -5.3999999999999997e-45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*l/ 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

   5. Simplified 100.0%

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

      1. add-sqr-sqrt 79.4%

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

      2. sqrt-unprod 61.1%

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

      3. sqr-neg 61.1%

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

      4. sqrt-unprod 20.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. expm1-log1p-u 77.8%

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

      7. expm1-udef 51.6%

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

      8. associate-*l/ 51.6%

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

   7. Applied egg-rr 51.6%

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

      1. expm1-def 77.5%

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

      2. expm1-log1p 99.7%

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

      3. associate-*l/ 100.0%

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

   9. Simplified 100.0%

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

   if -5.3999999999999997e-45 < x < 8.50000000000000006e-52

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 8.50000000000000006e-52 < x

   1. Initial program 90.0%

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

      1. associate-/r/ 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. clear-num 94.8%

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

      2. associate-/r/ 93.5%

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

      3. clear-num 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in x around inf 93.7%

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

      1. div-sub 93.7%

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

   9. Simplified 93.7%

      \[\leadsto \left|\color{blue}{x \cdot \frac{1 - z}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.3 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + 1\right)\right|\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-45}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{4 + x}{y_m}\\ t_1 := z \cdot \frac{x}{y_m}\\ \mathbf{if}\;t_0 - t_1 \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\left|t_1 - t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ 4.0 x) y_m)) (t_1 (* z (/ x y_m))))
   (if (<= (- t_0 t_1) -5e+106)
     (fabs (- t_1 t_0))
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (4.0 + x) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= -5e+106) {
		tmp = fabs((t_1 - t_0));
	} else {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 + x) / y_m
    t_1 = z * (x / y_m)
    if ((t_0 - t_1) <= (-5d+106)) then
        tmp = abs((t_1 - t_0))
    else
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = (4.0 + x) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= -5e+106) {
		tmp = Math.abs((t_1 - t_0));
	} else {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (4.0 + x) / y_m
	t_1 = z * (x / y_m)
	tmp = 0
	if (t_0 - t_1) <= -5e+106:
		tmp = math.fabs((t_1 - t_0))
	else:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(4.0 + x) / y_m)
	t_1 = Float64(z * Float64(x / y_m))
	tmp = 0.0
	if (Float64(t_0 - t_1) <= -5e+106)
		tmp = abs(Float64(t_1 - t_0));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (4.0 + x) / y_m;
	t_1 = z * (x / y_m);
	tmp = 0.0;
	if ((t_0 - t_1) <= -5e+106)
		tmp = abs((t_1 - t_0));
	else
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - t$95$1), $MachinePrecision], -5e+106], N[Abs[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < -4.9999999999999998e106

   1. Initial program 99.9%

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

   if -4.9999999999999998e106 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 91.4%

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

   3. Taylor expanded in y around 0 95.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - z \cdot \frac{x}{y} \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+163}:\\ \;\;\;\;\left|\frac{x}{y_m} \cdot \left(z + 1\right)\right|\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+17}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y_m} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1e+163)
   (fabs (* (/ x y_m) (+ z 1.0)))
   (if (<= x 6e+17)
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
     (fabs (* (/ x y_m) (- 1.0 z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1e+163) {
		tmp = fabs(((x / y_m) * (z + 1.0)));
	} else if (x <= 6e+17) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1d+163)) then
        tmp = abs(((x / y_m) * (z + 1.0d0)))
    else if (x <= 6d+17) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1e+163) {
		tmp = Math.abs(((x / y_m) * (z + 1.0)));
	} else if (x <= 6e+17) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1e+163:
		tmp = math.fabs(((x / y_m) * (z + 1.0)))
	elif x <= 6e+17:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1e+163)
		tmp = abs(Float64(Float64(x / y_m) * Float64(z + 1.0)));
	elseif (x <= 6e+17)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1e+163)
		tmp = abs(((x / y_m) * (z + 1.0)));
	elseif (x <= 6e+17)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs(((x / y_m) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1e+163], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6e+17], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999994e162

   1. Initial program 74.8%

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

   3. Taylor expanded in x around inf 74.8%

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

      1. sub-neg 74.8%

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

      2. distribute-rgt-neg-out 74.8%

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

      3. add-sqr-sqrt 60.5%

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

      4. sqrt-unprod 79.0%

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

      5. sqr-neg 79.0%

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

      6. sqrt-unprod 39.2%

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

      7. add-sqr-sqrt 85.5%

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

      8. associate-*l/ 69.3%

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

   5. Applied egg-rr 69.3%

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

      1. *-rgt-identity 69.3%

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

      2. associate-*l/ 85.5%

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

      3. distribute-lft-in 99.8%

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

   7. Simplified 99.8%

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

   if -9.9999999999999994e162 < x < 6e17

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0 99.9%

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

   if 6e17 < x

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 89.5%

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

      1. *-un-lft-identity 89.5%

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

      2. *-commutative 89.5%

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

      3. distribute-rgt-out-- 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+163}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + 1\right)\right|\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+17}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 6 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y_m} - x \cdot \frac{z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 6e-30)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (- (/ (+ 4.0 x) y_m) (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 6e-30) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 6d-30) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((((4.0d0 + x) / y_m) - (x * (z / y_m))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 6e-30) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 6e-30:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / y_m) - (x * (z / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 6e-30)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y_m) - Float64(x * Float64(z / y_m))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 6e-30)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((((4.0 + x) / y_m) - (x * (z / y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 6e-30], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.9999999999999998e-30

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 0 96.4%

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

   if 5.9999999999999998e-30 < y

   1. Initial program 96.8%

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

      1. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+101}:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+92}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.45e+101)
   (fabs (/ z (/ y_m x)))
   (if (<= z 2.25e+92) (fabs (/ (- -4.0 x) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.45e+101) {
		tmp = fabs((z / (y_m / x)));
	} else if (z <= 2.25e+92) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.45d+101)) then
        tmp = abs((z / (y_m / x)))
    else if (z <= 2.25d+92) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.45e+101) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (z <= 2.25e+92) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.45e+101:
		tmp = math.fabs((z / (y_m / x)))
	elif z <= 2.25e+92:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.45e+101)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (z <= 2.25e+92)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.45e+101)
		tmp = abs((z / (y_m / x)));
	elseif (z <= 2.25e+92)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.45e+101], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.25e+92], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.44999999999999994e101

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf 69.2%

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

      1. mul-1-neg 69.2%

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

      2. associate-*l/ 77.2%

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

      3. distribute-rgt-neg-out 77.2%

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

   5. Simplified 77.2%

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

      1. add-sqr-sqrt 77.0%

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

      2. clear-num 77.0%

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

      3. sqrt-unprod 51.2%

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

      4. sqr-neg 51.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 77.1%

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

      7. associate-*l/ 77.2%

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

      8. *-un-lft-identity 77.2%

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

   7. Applied egg-rr 77.2%

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

   if -1.44999999999999994e101 < z < 2.25e92

   1. Initial program 94.4%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in z around 0 88.9%

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

      1. associate-*r/ 88.9%

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

      2. distribute-lft-in 88.9%

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

      3. metadata-eval 88.9%

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

      4. neg-mul-1 88.9%

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

      5. sub-neg 88.9%

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

   7. Simplified 88.9%

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

   if 2.25e92 < z

   1. Initial program 84.5%

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

   3. Taylor expanded in z around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. associate-*l/ 73.2%

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

      3. distribute-rgt-neg-out 73.2%

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

   5. Simplified 73.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 47.0%

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

      3. sqr-neg 47.0%

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

      4. sqrt-unprod 72.9%

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

      5. add-sqr-sqrt 73.2%

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

      6. expm1-log1p-u 32.1%

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

      7. expm1-udef 21.5%

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

      8. associate-*l/ 19.5%

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

   7. Applied egg-rr 19.5%

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

      1. expm1-def 33.0%

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

      2. expm1-log1p 66.7%

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

      3. associate-*l/ 73.2%

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

   9. Simplified 73.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+101}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+92}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.5) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 4 < x

   1. Initial program 86.9%

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

   3. Taylor expanded in z around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in x around inf 59.2%

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

   if -1.5 < x < 4

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0 78.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in x around 0 47.9%

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

4. Final simplification 47.9%

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))