fabs fraction 1

Percentage Accurate: 91.8% → 99.7%

Time: 7.7s

Alternatives: 10

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}} - \frac{4 + x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4.2e-47)
   (fabs (/ (+ 4.0 (* x (- 1.0 z))) y_m))
   (fabs (- (/ x (/ y_m z)) (/ (+ 4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.2e-47) {
		tmp = fabs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = fabs(((x / (y_m / z)) - ((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 (y_m <= 4.2d-47) then
        tmp = abs(((4.0d0 + (x * (1.0d0 - z))) / y_m))
    else
        tmp = abs(((x / (y_m / z)) - ((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 (y_m <= 4.2e-47) {
		tmp = Math.abs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = Math.abs(((x / (y_m / z)) - ((4.0 + x) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 4.2e-47:
		tmp = math.fabs(((4.0 + (x * (1.0 - z))) / y_m))
	else:
		tmp = math.fabs(((x / (y_m / z)) - ((4.0 + x) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.2e-47)
		tmp = abs(Float64(Float64(4.0 + Float64(x * Float64(1.0 - z))) / y_m));
	else
		tmp = abs(Float64(Float64(x / Float64(y_m / z)) - 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 (y_m <= 4.2e-47)
		tmp = abs(((4.0 + (x * (1.0 - z))) / y_m));
	else
		tmp = abs(((x / (y_m / z)) - ((4.0 + 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, 4.2e-47], N[Abs[N[(N[(4.0 + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000001e-47

   1. Initial program 92.3%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.9%

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

      1. associate-*r/ 98.9%

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

      2. associate--r+ 98.9%

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

      3. sub-neg 98.9%

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

      4. metadata-eval 98.9%

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

      5. associate-+r- 98.9%

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

      6. fma-undefine 98.9%

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

      7. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   if 4.2000000000000001e-47 < y

   1. Initial program 95.1%

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

      1. associate-*l/ 93.9%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{4 + x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+128} \lor \neg \left(x \leq 9.5 \cdot 10^{+174}\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))))
   (if (<= x -2.15e-47)
     t_0
     (if (<= x 4.0)
       (fabs (/ 4.0 y_m))
       (if (or (<= x 6.2e+128) (not (<= x 9.5e+174))) (fabs (/ x y_m)) 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 tmp;
	if (x <= -2.15e-47) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 6.2e+128) || !(x <= 9.5e+174)) {
		tmp = fabs((x / 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((z * (x / y_m)))
    if (x <= (-2.15d-47)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 6.2d+128) .or. (.not. (x <= 9.5d+174))) then
        tmp = abs((x / 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((z * (x / y_m)));
	double tmp;
	if (x <= -2.15e-47) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 6.2e+128) || !(x <= 9.5e+174)) {
		tmp = Math.abs((x / y_m));
	} 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)))
	tmp = 0
	if x <= -2.15e-47:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 6.2e+128) or not (x <= 9.5e+174):
		tmp = math.fabs((x / y_m))
	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)))
	tmp = 0.0
	if (x <= -2.15e-47)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 6.2e+128) || !(x <= 9.5e+174))
		tmp = abs(Float64(x / 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((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -2.15e-47)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 6.2e+128) || ~((x <= 9.5e+174)))
		tmp = abs((x / 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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.15e-47], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 6.2e+128], N[Not[LessEqual[x, 9.5e+174]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1499999999999999e-47 or 6.20000000000000008e128 < x < 9.4999999999999992e174

   1. Initial program 94.1%

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

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 57.7%

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

      1. mul-1-neg 57.7%

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

      2. distribute-frac-neg2 57.7%

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

      3. associate-/l* 63.0%

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

   7. Simplified 63.0%

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

      1. clear-num 63.0%

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

      2. un-div-inv 63.1%

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

      3. add-sqr-sqrt 33.6%

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

      4. sqrt-unprod 59.9%

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

      5. sqr-neg 59.9%

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

      6. sqrt-unprod 29.4%

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

      7. add-sqr-sqrt 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. associate-/r/ 70.6%

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

       2. *-commutative 70.6%

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

   11. Simplified 70.6%

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

   if -2.1499999999999999e-47 < x < 4

   1. Initial program 94.4%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 74.3%

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

   if 4 < x < 6.20000000000000008e128 or 9.4999999999999992e174 < x

   1. Initial program 89.4%

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

   3. Simplified 96.5%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 92.5%

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

      1. mul-1-neg 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-/l* 95.8%

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

      4. distribute-lft-neg-in 95.8%

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

      5. neg-sub0 95.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 95.8%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 95.8%

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

      8. +-commutative 95.8%

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

      9. unsub-neg 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in z around 0 73.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-47}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+128} \lor \neg \left(x \leq 9.5 \cdot 10^{+174}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 4.2)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (- 4.0 (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 4.2)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs(((4.0 - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 4.2d0))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs(((4.0d0 - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 4.2)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs(((4.0 - (x * z)) / 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 <= 4.2):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs(((4.0 - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 4.2))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(4.0 - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 4.2)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs(((4.0 - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 4.2]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 4.20000000000000018 < x

   1. Initial program 91.9%

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

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. *-commutative 93.0%

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

      3. associate-/l* 97.8%

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

      4. distribute-lft-neg-in 97.8%

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

      5. neg-sub0 97.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 97.8%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 97.8%

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

      8. +-commutative 97.8%

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

      9. unsub-neg 97.8%

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

   7. Simplified 97.8%

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

   if -1.5 < x < 4.20000000000000018

   1. Initial program 94.7%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate--r+ 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+r- 99.9%

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

      6. fma-undefine 99.9%

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

      7. neg-mul-1 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf 99.0%

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

      1. associate-*r* 99.0%

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

      2. neg-mul-1 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -350000:\\ \;\;\;\;\left|\frac{x \cdot \left(1 - z\right)}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 + z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -350000.0)
   (fabs (/ (* x (- 1.0 z)) y_m))
   (if (<= z 2.5e+19)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (/ (+ 1.0 z) (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -350000.0) {
		tmp = fabs(((x * (1.0 - z)) / y_m));
	} else if (z <= 2.5e+19) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((1.0 + z) / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-350000.0d0)) then
        tmp = abs(((x * (1.0d0 - z)) / y_m))
    else if (z <= 2.5d+19) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((1.0d0 + z) / (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -350000.0) {
		tmp = Math.abs(((x * (1.0 - z)) / y_m));
	} else if (z <= 2.5e+19) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((1.0 + z) / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -350000.0:
		tmp = math.fabs(((x * (1.0 - z)) / y_m))
	elif z <= 2.5e+19:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((1.0 + z) / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -350000.0)
		tmp = abs(Float64(Float64(x * Float64(1.0 - z)) / y_m));
	elseif (z <= 2.5e+19)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 + z) / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -350000.0)
		tmp = abs(((x * (1.0 - z)) / y_m));
	elseif (z <= 2.5e+19)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((1.0 + z) / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -350000.0], N[Abs[N[(N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+19], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 + z), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5e5

   1. Initial program 92.5%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. associate--r+ 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

      5. associate-+r- 98.4%

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

      6. fma-undefine 98.4%

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

      7. neg-mul-1 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around inf 75.4%

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

   if -3.5e5 < z < 2.5e19

   1. Initial program 95.5%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. rem-square-sqrt 49.2%

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

      3. fabs-sqr 49.2%

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

      4. rem-square-sqrt 98.3%

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

      5. fabs-neg 98.3%

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

      6. distribute-neg-frac 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. sub-neg 98.3%

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

      11. rem-square-sqrt 48.5%

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

      12. fabs-sqr 48.5%

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

      13. rem-square-sqrt 98.3%

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

   7. Simplified 98.3%

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

   if 2.5e19 < z

   1. Initial program 88.8%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-/l* 80.9%

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

      4. distribute-lft-neg-in 80.9%

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

      5. neg-sub0 80.9%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 80.9%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 80.9%

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

      8. +-commutative 80.9%

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

      9. unsub-neg 80.9%

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

   7. Simplified 80.9%

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

      1. clear-num 80.9%

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

      2. un-div-inv 80.9%

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

      3. sub-neg 80.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 54.3%

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

      6. sqr-neg 54.3%

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

      7. sqrt-unprod 80.6%

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

      8. add-sqr-sqrt 80.9%

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

   9. Applied egg-rr 80.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -350000:\\ \;\;\;\;\left|\frac{x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 + z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{\left(z + -1\right) - \frac{4}{x}}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e+24)
   (fabs (/ (+ 4.0 (* x (- 1.0 z))) y_m))
   (fabs (* x (/ (- (+ z -1.0) (/ 4.0 x)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e+24) {
		tmp = fabs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = fabs((x * (((z + -1.0) - (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 (y_m <= 2d+24) then
        tmp = abs(((4.0d0 + (x * (1.0d0 - z))) / y_m))
    else
        tmp = abs((x * (((z + (-1.0d0)) - (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 (y_m <= 2e+24) {
		tmp = Math.abs(((4.0 + (x * (1.0 - z))) / y_m));
	} else {
		tmp = Math.abs((x * (((z + -1.0) - (4.0 / x)) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 2e+24:
		tmp = math.fabs(((4.0 + (x * (1.0 - z))) / y_m))
	else:
		tmp = math.fabs((x * (((z + -1.0) - (4.0 / x)) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e+24)
		tmp = abs(Float64(Float64(4.0 + Float64(x * Float64(1.0 - z))) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(Float64(z + -1.0) - 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 (y_m <= 2e+24)
		tmp = abs(((4.0 + (x * (1.0 - z))) / y_m));
	else
		tmp = abs((x * (((z + -1.0) - (4.0 / 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, 2e+24], N[Abs[N[(N[(4.0 + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(N[(z + -1.0), $MachinePrecision] - N[(4.0 / x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e24

   1. Initial program 93.0%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. associate--r+ 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. associate-+r- 99.0%

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

      6. fma-undefine 99.0%

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

      7. neg-mul-1 99.0%

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

   7. Simplified 99.0%

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

   if 2e24 < y

   1. Initial program 93.9%

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

      1. fabs-sub 93.9%

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

      2. associate-*l/ 92.3%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.7%

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

      1. associate--r+ 99.7%

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

      2. div-sub 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. +-commutative 99.7%

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

      6. metadata-eval 99.7%

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

      7. remove-double-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. neg-mul-1 99.7%

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

      11. associate-*r/ 99.7%

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

   7. Simplified 99.7%

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

      1. associate-/r* 99.7%

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

      2. sub-div 99.7%

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

   9. Applied egg-rr 99.7%

      \[\leadsto \left|x \cdot \color{blue}{\frac{\left(z + -1\right) - \frac{4}{x}}{y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{\left(z + -1\right) - \frac{4}{x}}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -380000:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -380000.0)
   (fabs (/ (* x z) y_m))
   (if (<= z 2.5e+19) (fabs (/ (- -4.0 x) y_m)) (fabs (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -380000.0) {
		tmp = fabs(((x * z) / y_m));
	} else if (z <= 2.5e+19) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-380000.0d0)) then
        tmp = abs(((x * z) / y_m))
    else if (z <= 2.5d+19) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z / (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -380000.0) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (z <= 2.5e+19) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -380000.0:
		tmp = math.fabs(((x * z) / y_m))
	elif z <= 2.5e+19:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -380000.0)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (z <= 2.5e+19)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -380000.0)
		tmp = abs(((x * z) / y_m));
	elseif (z <= 2.5e+19)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -380000.0], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+19], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8e5

   1. Initial program 92.5%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. distribute-frac-neg2 74.6%

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

      3. associate-/l* 73.7%

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

   7. Simplified 73.7%

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

      1. associate-*r/ 74.6%

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

      2. add-sqr-sqrt 34.6%

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

      3. sqrt-unprod 57.8%

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

      4. sqr-neg 57.8%

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

      5. sqrt-unprod 39.9%

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

      6. add-sqr-sqrt 74.6%

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

   9. Applied egg-rr 74.6%

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

   if -3.8e5 < z < 2.5e19

   1. Initial program 95.5%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. rem-square-sqrt 49.2%

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

      3. fabs-sqr 49.2%

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

      4. rem-square-sqrt 98.3%

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

      5. fabs-neg 98.3%

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

      6. distribute-neg-frac 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. sub-neg 98.3%

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

      11. rem-square-sqrt 48.5%

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

      12. fabs-sqr 48.5%

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

      13. rem-square-sqrt 98.3%

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

   7. Simplified 98.3%

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

   if 2.5e19 < z

   1. Initial program 88.8%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. distribute-frac-neg2 73.2%

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

      3. associate-/l* 76.3%

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

   7. Simplified 76.3%

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

      1. clear-num 76.3%

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

      2. un-div-inv 77.1%

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

      3. add-sqr-sqrt 32.4%

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

      4. sqrt-unprod 54.4%

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

      5. sqr-neg 54.4%

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

      6. sqrt-unprod 44.5%

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

      7. add-sqr-sqrt 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. associate-/r/ 80.9%

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

       2. *-commutative 80.9%

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

   11. Simplified 80.9%

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

       1. clear-num 80.9%

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

       2. un-div-inv 80.9%

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

   13. Applied egg-rr 80.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -380000:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;\left|\frac{x \cdot \left(1 - z\right)}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -320000.0)
   (fabs (/ (* x (- 1.0 z)) y_m))
   (if (<= z 1.15e+20) (fabs (/ (- -4.0 x) y_m)) (fabs (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -320000.0) {
		tmp = fabs(((x * (1.0 - z)) / y_m));
	} else if (z <= 1.15e+20) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-320000.0d0)) then
        tmp = abs(((x * (1.0d0 - z)) / y_m))
    else if (z <= 1.15d+20) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z / (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -320000.0) {
		tmp = Math.abs(((x * (1.0 - z)) / y_m));
	} else if (z <= 1.15e+20) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -320000.0:
		tmp = math.fabs(((x * (1.0 - z)) / y_m))
	elif z <= 1.15e+20:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -320000.0)
		tmp = abs(Float64(Float64(x * Float64(1.0 - z)) / y_m));
	elseif (z <= 1.15e+20)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -320000.0)
		tmp = abs(((x * (1.0 - z)) / y_m));
	elseif (z <= 1.15e+20)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -320000.0], N[Abs[N[(N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.15e+20], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2e5

   1. Initial program 92.5%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. associate--r+ 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

      5. associate-+r- 98.4%

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

      6. fma-undefine 98.4%

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

      7. neg-mul-1 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around inf 75.4%

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

   if -3.2e5 < z < 1.15e20

   1. Initial program 95.5%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. rem-square-sqrt 49.2%

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

      3. fabs-sqr 49.2%

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

      4. rem-square-sqrt 98.3%

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

      5. fabs-neg 98.3%

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

      6. distribute-neg-frac 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. sub-neg 98.3%

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

      11. rem-square-sqrt 48.5%

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

      12. fabs-sqr 48.5%

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

      13. rem-square-sqrt 98.3%

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

   7. Simplified 98.3%

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

   if 1.15e20 < z

   1. Initial program 88.8%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. distribute-frac-neg2 73.2%

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

      3. associate-/l* 76.3%

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

   7. Simplified 76.3%

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

      1. clear-num 76.3%

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

      2. un-div-inv 77.1%

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

      3. add-sqr-sqrt 32.4%

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

      4. sqrt-unprod 54.4%

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

      5. sqr-neg 54.4%

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

      6. sqrt-unprod 44.5%

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

      7. add-sqr-sqrt 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. associate-/r/ 80.9%

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

       2. *-commutative 80.9%

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

   11. Simplified 80.9%

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

       1. clear-num 80.9%

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

       2. un-div-inv 80.9%

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

   13. Applied egg-rr 80.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -320000:\\ \;\;\;\;\left|\frac{x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% 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 91.9%

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

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. *-commutative 93.0%

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

      3. associate-/l* 97.8%

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

      4. distribute-lft-neg-in 97.8%

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

      5. neg-sub0 97.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 97.8%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 97.8%

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

      8. +-commutative 97.8%

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

      9. unsub-neg 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in z around 0 65.7%

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

   if -1.5 < x < 4

   1. Initial program 94.7%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.1%

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

4. Final simplification 68.7%

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

Alternative 9: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{4 + x \cdot \left(1 - z\right)}{y\_m}\right| \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (+ 4.0 (* x (- 1.0 z))) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs(((4.0 + (x * (1.0 - 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(((4.0d0 + (x * (1.0d0 - 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(((4.0 + (x * (1.0 - z))) / y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return math.fabs(((4.0 + (x * (1.0 - z))) / y_m))

Julia

y_m = abs(y)
function code(x, y_m, z)
	return abs(Float64(Float64(4.0 + Float64(x * Float64(1.0 - z))) / y_m))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = abs(((4.0 + (x * (1.0 - z))) / y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(4.0 + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 93.2%

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

3. Simplified 97.3%

   \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 97.4%

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

   1. associate-*r/ 97.4%

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

   2. associate--r+ 97.4%

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

   3. sub-neg 97.4%

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

   4. metadata-eval 97.4%

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

   5. associate-+r- 97.4%

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

   6. fma-undefine 97.4%

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

   7. neg-mul-1 97.4%

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

7. Simplified 97.4%

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

8. Final simplification 97.4%

   \[\leadsto \left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right| \]
9. Add Preprocessing

Alternative 10: 41.0% 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 93.2%

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

3. Simplified 97.3%

   \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 36.4%

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

6. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2024075 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))