fabs fraction 1

Percentage Accurate: 91.9% → 99.8%

Time: 7.1s

Alternatives: 9

Speedup: 1.0×

• 20.1% 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: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 8e-47)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8e-47) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 8e-47)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 8e-47], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999998e-47

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0 97.5%

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

   if 7.9999999999999998e-47 < y

   1. Initial program 98.1%

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

   3. Simplified 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{4}{y\_m}\right|\\ t_1 := \left|\frac{x}{y\_m}\right|\\ t_2 := \left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-90}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+98}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y_m)))
        (t_1 (fabs (/ x y_m)))
        (t_2 (fabs (* x (/ z y_m)))))
   (if (<= x -2.6e+104)
     t_1
     (if (<= x -1e-16)
       t_2
       (if (<= x -6.2e-54)
         t_0
         (if (<= x -1.1e-90)
           t_2
           (if (<= x 3.6e-22) t_0 (if (<= x 3.2e+98) t_2 t_1))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((4.0 / y_m));
	double t_1 = fabs((x / y_m));
	double t_2 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -2.6e+104) {
		tmp = t_1;
	} else if (x <= -1e-16) {
		tmp = t_2;
	} else if (x <= -6.2e-54) {
		tmp = t_0;
	} else if (x <= -1.1e-90) {
		tmp = t_2;
	} else if (x <= 3.6e-22) {
		tmp = t_0;
	} else if (x <= 3.2e+98) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = abs((4.0d0 / y_m))
    t_1 = abs((x / y_m))
    t_2 = abs((x * (z / y_m)))
    if (x <= (-2.6d+104)) then
        tmp = t_1
    else if (x <= (-1d-16)) then
        tmp = t_2
    else if (x <= (-6.2d-54)) then
        tmp = t_0
    else if (x <= (-1.1d-90)) then
        tmp = t_2
    else if (x <= 3.6d-22) then
        tmp = t_0
    else if (x <= 3.2d+98) then
        tmp = t_2
    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((4.0 / y_m));
	double t_1 = Math.abs((x / y_m));
	double t_2 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -2.6e+104) {
		tmp = t_1;
	} else if (x <= -1e-16) {
		tmp = t_2;
	} else if (x <= -6.2e-54) {
		tmp = t_0;
	} else if (x <= -1.1e-90) {
		tmp = t_2;
	} else if (x <= 3.6e-22) {
		tmp = t_0;
	} else if (x <= 3.2e+98) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((4.0 / y_m))
	t_1 = math.fabs((x / y_m))
	t_2 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -2.6e+104:
		tmp = t_1
	elif x <= -1e-16:
		tmp = t_2
	elif x <= -6.2e-54:
		tmp = t_0
	elif x <= -1.1e-90:
		tmp = t_2
	elif x <= 3.6e-22:
		tmp = t_0
	elif x <= 3.2e+98:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(4.0 / y_m))
	t_1 = abs(Float64(x / y_m))
	t_2 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -2.6e+104)
		tmp = t_1;
	elseif (x <= -1e-16)
		tmp = t_2;
	elseif (x <= -6.2e-54)
		tmp = t_0;
	elseif (x <= -1.1e-90)
		tmp = t_2;
	elseif (x <= 3.6e-22)
		tmp = t_0;
	elseif (x <= 3.2e+98)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((4.0 / y_m));
	t_1 = abs((x / y_m));
	t_2 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -2.6e+104)
		tmp = t_1;
	elseif (x <= -1e-16)
		tmp = t_2;
	elseif (x <= -6.2e-54)
		tmp = t_0;
	elseif (x <= -1.1e-90)
		tmp = t_2;
	elseif (x <= 3.6e-22)
		tmp = t_0;
	elseif (x <= 3.2e+98)
		tmp = t_2;
	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[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.6e+104], t$95$1, If[LessEqual[x, -1e-16], t$95$2, If[LessEqual[x, -6.2e-54], t$95$0, If[LessEqual[x, -1.1e-90], t$95$2, If[LessEqual[x, 3.6e-22], t$95$0, If[LessEqual[x, 3.2e+98], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e104 or 3.2000000000000002e98 < x

   1. Initial program 85.3%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around inf 99.9%

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

   6. Taylor expanded in z around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. neg-mul-1 76.2%

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

   8. Simplified 76.2%

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

   if -2.6e104 < x < -9.9999999999999998e-17 or -6.20000000000000008e-54 < x < -1.09999999999999993e-90 or 3.5999999999999998e-22 < x < 3.2000000000000002e98

   1. Initial program 93.3%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 95.7%

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

   6. Taylor expanded in z around inf 66.4%

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

      1. associate-*r/ 71.0%

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

   8. Simplified 71.0%

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

   if -9.9999999999999998e-17 < x < -6.20000000000000008e-54 or -1.09999999999999993e-90 < x < 3.5999999999999998e-22

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 80.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+104}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-90}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+98}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ t_1 := \left|\frac{4}{y\_m}\right|\\ \mathbf{if}\;x \leq -7 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-90}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ 4.0 y_m))))
   (if (<= x -7e-12)
     t_0
     (if (<= x -1.4e-55)
       t_1
       (if (<= x -9.5e-90)
         (fabs (* x (/ z y_m)))
         (if (<= x 3.5e-22) t_1 (if (<= x 8e+101) t_0 (fabs (/ x y_m)))))))))

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((4.0 / y_m));
	double tmp;
	if (x <= -7e-12) {
		tmp = t_0;
	} else if (x <= -1.4e-55) {
		tmp = t_1;
	} else if (x <= -9.5e-90) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 3.5e-22) {
		tmp = t_1;
	} else if (x <= 8e+101) {
		tmp = t_0;
	} else {
		tmp = fabs((x / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((4.0d0 / y_m))
    if (x <= (-7d-12)) then
        tmp = t_0
    else if (x <= (-1.4d-55)) then
        tmp = t_1
    else if (x <= (-9.5d-90)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 3.5d-22) then
        tmp = t_1
    else if (x <= 8d+101) then
        tmp = t_0
    else
        tmp = abs((x / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((4.0 / y_m));
	double tmp;
	if (x <= -7e-12) {
		tmp = t_0;
	} else if (x <= -1.4e-55) {
		tmp = t_1;
	} else if (x <= -9.5e-90) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 3.5e-22) {
		tmp = t_1;
	} else if (x <= 8e+101) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y_m));
	}
	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((4.0 / y_m))
	tmp = 0
	if x <= -7e-12:
		tmp = t_0
	elif x <= -1.4e-55:
		tmp = t_1
	elif x <= -9.5e-90:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 3.5e-22:
		tmp = t_1
	elif x <= 8e+101:
		tmp = t_0
	else:
		tmp = math.fabs((x / y_m))
	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(4.0 / y_m))
	tmp = 0.0
	if (x <= -7e-12)
		tmp = t_0;
	elseif (x <= -1.4e-55)
		tmp = t_1;
	elseif (x <= -9.5e-90)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 3.5e-22)
		tmp = t_1;
	elseif (x <= 8e+101)
		tmp = t_0;
	else
		tmp = abs(Float64(x / y_m));
	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((4.0 / y_m));
	tmp = 0.0;
	if (x <= -7e-12)
		tmp = t_0;
	elseif (x <= -1.4e-55)
		tmp = t_1;
	elseif (x <= -9.5e-90)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 3.5e-22)
		tmp = t_1;
	elseif (x <= 8e+101)
		tmp = t_0;
	else
		tmp = abs((x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7e-12], t$95$0, If[LessEqual[x, -1.4e-55], t$95$1, If[LessEqual[x, -9.5e-90], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.5e-22], t$95$1, If[LessEqual[x, 8e+101], t$95$0, N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.0000000000000001e-12 or 3.50000000000000005e-22 < x < 7.9999999999999998e101

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf 58.4%

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

      1. mul-1-neg 58.4%

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

      2. associate-*l/ 69.2%

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

      3. distribute-rgt-neg-out 69.2%

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

   5. Simplified 69.2%

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

      1. add-log-exp 42.8%

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

      2. exp-prod 41.6%

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

      3. add-sqr-sqrt 20.8%

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

      4. sqrt-unprod 36.0%

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

      5. sqr-neg 36.0%

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

      6. sqrt-unprod 22.1%

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

      7. add-sqr-sqrt 41.6%

         \[\leadsto \left|\log \left({\left(e^{\frac{x}{y}}\right)}^{\color{blue}{z}}\right)\right| \]

   7. Applied egg-rr 41.6%

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

      1. log-pow 42.1%

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

      2. rem-log-exp 69.2%

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

   9. Simplified 69.2%

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

   if -7.0000000000000001e-12 < x < -1.39999999999999992e-55 or -9.5000000000000003e-90 < x < 3.50000000000000005e-22

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 80.0%

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

   if -1.39999999999999992e-55 < x < -9.5000000000000003e-90

   1. Initial program 99.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 88.2%

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

   6. Taylor expanded in z around inf 88.2%

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

      1. associate-*r/ 88.2%

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

   8. Simplified 88.2%

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

   if 7.9999999999999998e101 < x

   1. Initial program 82.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around inf 99.9%

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

   6. Taylor expanded in z around 0 83.6%

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

      1. associate-*r/ 83.6%

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

      2. neg-mul-1 83.6%

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

   8. Simplified 83.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-12}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-90}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+101}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ t_1 := \left|\frac{4}{y\_m}\right|\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+107}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ 4.0 y_m))))
   (if (<= x -8.6e-15)
     t_0
     (if (<= x -1.15e-51)
       t_1
       (if (<= x -1.3e-89)
         (fabs (/ z (/ y_m x)))
         (if (<= x 3.1e-22) t_1 (if (<= x 3.2e+107) t_0 (fabs (/ x y_m)))))))))

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((4.0 / y_m));
	double tmp;
	if (x <= -8.6e-15) {
		tmp = t_0;
	} else if (x <= -1.15e-51) {
		tmp = t_1;
	} else if (x <= -1.3e-89) {
		tmp = fabs((z / (y_m / x)));
	} else if (x <= 3.1e-22) {
		tmp = t_1;
	} else if (x <= 3.2e+107) {
		tmp = t_0;
	} else {
		tmp = fabs((x / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((4.0d0 / y_m))
    if (x <= (-8.6d-15)) then
        tmp = t_0
    else if (x <= (-1.15d-51)) then
        tmp = t_1
    else if (x <= (-1.3d-89)) then
        tmp = abs((z / (y_m / x)))
    else if (x <= 3.1d-22) then
        tmp = t_1
    else if (x <= 3.2d+107) then
        tmp = t_0
    else
        tmp = abs((x / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((4.0 / y_m));
	double tmp;
	if (x <= -8.6e-15) {
		tmp = t_0;
	} else if (x <= -1.15e-51) {
		tmp = t_1;
	} else if (x <= -1.3e-89) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (x <= 3.1e-22) {
		tmp = t_1;
	} else if (x <= 3.2e+107) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y_m));
	}
	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((4.0 / y_m))
	tmp = 0
	if x <= -8.6e-15:
		tmp = t_0
	elif x <= -1.15e-51:
		tmp = t_1
	elif x <= -1.3e-89:
		tmp = math.fabs((z / (y_m / x)))
	elif x <= 3.1e-22:
		tmp = t_1
	elif x <= 3.2e+107:
		tmp = t_0
	else:
		tmp = math.fabs((x / y_m))
	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(4.0 / y_m))
	tmp = 0.0
	if (x <= -8.6e-15)
		tmp = t_0;
	elseif (x <= -1.15e-51)
		tmp = t_1;
	elseif (x <= -1.3e-89)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (x <= 3.1e-22)
		tmp = t_1;
	elseif (x <= 3.2e+107)
		tmp = t_0;
	else
		tmp = abs(Float64(x / y_m));
	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((4.0 / y_m));
	tmp = 0.0;
	if (x <= -8.6e-15)
		tmp = t_0;
	elseif (x <= -1.15e-51)
		tmp = t_1;
	elseif (x <= -1.3e-89)
		tmp = abs((z / (y_m / x)));
	elseif (x <= 3.1e-22)
		tmp = t_1;
	elseif (x <= 3.2e+107)
		tmp = t_0;
	else
		tmp = abs((x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8.6e-15], t$95$0, If[LessEqual[x, -1.15e-51], t$95$1, If[LessEqual[x, -1.3e-89], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.1e-22], t$95$1, If[LessEqual[x, 3.2e+107], t$95$0, N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.5999999999999993e-15 or 3.10000000000000013e-22 < x < 3.20000000000000029e107

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf 58.4%

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

      1. mul-1-neg 58.4%

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

      2. associate-*l/ 69.2%

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

      3. distribute-rgt-neg-out 69.2%

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

   5. Simplified 69.2%

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

      1. add-log-exp 42.8%

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

      2. exp-prod 41.6%

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

      3. add-sqr-sqrt 20.8%

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

      4. sqrt-unprod 36.0%

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

      5. sqr-neg 36.0%

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

      6. sqrt-unprod 22.1%

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

      7. add-sqr-sqrt 41.6%

         \[\leadsto \left|\log \left({\left(e^{\frac{x}{y}}\right)}^{\color{blue}{z}}\right)\right| \]

   7. Applied egg-rr 41.6%

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

      1. log-pow 42.1%

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

      2. rem-log-exp 69.2%

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

   9. Simplified 69.2%

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

   if -8.5999999999999993e-15 < x < -1.15000000000000001e-51 or -1.2999999999999999e-89 < x < 3.10000000000000013e-22

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 80.0%

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

   if -1.15000000000000001e-51 < x < -1.2999999999999999e-89

   1. Initial program 99.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 88.2%

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

   6. Taylor expanded in z around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. *-commutative 88.2%

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

      3. associate-/r/ 88.2%

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

   8. Simplified 88.2%

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

   if 3.20000000000000029e107 < x

   1. Initial program 82.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around inf 99.9%

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

   6. Taylor expanded in z around 0 83.6%

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

      1. associate-*r/ 83.6%

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

      2. neg-mul-1 83.6%

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

   8. Simplified 83.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-15}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-22}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+107}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 7e-81)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 7e-81) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (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 (y_m <= 7d-81) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (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 (y_m <= 7e-81) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 7e-81:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (z * (x / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 7e-81)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(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 (y_m <= 7e-81)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (z * (x / 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, 7e-81], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999973e-81

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0 97.4%

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

   if 6.99999999999999973e-81 < y

   1. Initial program 98.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+38} \lor \neg \left(z \leq 1.75 \cdot 10^{+130}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.05e+38) (not (<= z 1.75e+130)))
   (fabs (* z (/ x y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.05e+38) || !(z <= 1.75e+130)) {
		tmp = fabs((z * (x / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.05d+38)) .or. (.not. (z <= 1.75d+130))) then
        tmp = abs((z * (x / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.05e+38) || !(z <= 1.75e+130)) {
		tmp = Math.abs((z * (x / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.05e+38) or not (z <= 1.75e+130):
		tmp = math.fabs((z * (x / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.05e+38) || !(z <= 1.75e+130))
		tmp = abs(Float64(z * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.05e+38) || ~((z <= 1.75e+130)))
		tmp = abs((z * (x / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.05e+38], N[Not[LessEqual[z, 1.75e+130]], $MachinePrecision]], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e38 or 1.75e130 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in z around inf 78.6%

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

      1. mul-1-neg 78.6%

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

      2. associate-*l/ 83.8%

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

      3. distribute-rgt-neg-out 83.8%

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

   5. Simplified 83.8%

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

      1. add-log-exp 38.8%

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

      2. exp-prod 35.3%

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

      3. add-sqr-sqrt 23.4%

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

      4. sqrt-unprod 35.3%

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

      5. sqr-neg 35.3%

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

      6. sqrt-unprod 13.6%

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

      7. add-sqr-sqrt 35.3%

         \[\leadsto \left|\log \left({\left(e^{\frac{x}{y}}\right)}^{\color{blue}{z}}\right)\right| \]

   7. Applied egg-rr 35.3%

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

      1. log-pow 36.9%

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

      2. rem-log-exp 83.8%

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

   9. Simplified 83.8%

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

   if -1.05e38 < z < 1.75e130

   1. Initial program 90.6%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in z around 0 93.2%

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

      1. associate-*r/ 93.2%

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

      2. distribute-lft-in 93.2%

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

      3. metadata-eval 93.2%

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

      4. neg-mul-1 93.2%

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

      5. sub-neg 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+38} \lor \neg \left(z \leq 1.75 \cdot 10^{+130}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% 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 87.4%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around inf 99.2%

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

   6. Taylor expanded in z around 0 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   8. Simplified 64.3%

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

   if -1.5 < x < 4

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0 72.3%

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

4. Final simplification 68.3%

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

Alternative 8: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs((((x + 4.0) - (x * z)) / 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((((x + 4.0d0) - (x * z)) / y_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in y around 0 97.4%

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

4. Final simplification 97.4%

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

Alternative 9: 40.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.8%

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

3. Taylor expanded in x around 0 38.7%

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

4. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2024027 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))