fabs fraction 1

Percentage Accurate: 91.7% → 99.9%

Time: 7.2s

Alternatives: 11

Speedup: 1.0×

• 20.4% 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: 91.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-16)
   (fabs (/ (- (* x z) (+ x 4.0)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-16) {
		tmp = fabs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 4e-16)
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 4e-16], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999999e-16

   1. Initial program 92.3%

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

      1. associate-*l/ 95.0%

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

      2. associate-*r/ 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around 0 96.3%

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

      1. sub-neg 96.3%

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

      2. +-commutative 96.3%

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

      3. distribute-lft-in 92.4%

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

      4. associate-+r+ 92.4%

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

      5. distribute-rgt-in 92.4%

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

      6. associate-*l/ 92.5%

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

      7. *-lft-identity 92.5%

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

      8. +-commutative 92.5%

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

      9. distribute-rgt-neg-out 92.5%

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

      10. sub-neg 92.5%

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

      11. associate-*r/ 95.0%

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

      12. div-sub 98.9%

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

   6. Simplified 98.9%

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

   if 3.9999999999999999e-16 < y

   1. Initial program 93.0%

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

   3. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+107}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+127} \lor \neg \left(x \leq 6.2 \cdot 10^{+178}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (/ x (/ y z)))))
   (if (<= x -1.35e+32)
     t_0
     (if (<= x -1.95e-94)
       t_1
       (if (<= x 1.35e-10)
         (fabs (/ 4.0 y))
         (if (<= x 3.3e+107)
           (fabs (* z (/ x y)))
           (if (or (<= x 3.4e+127) (not (<= x 6.2e+178))) t_0 t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((x / (y / z)));
	double tmp;
	if (x <= -1.35e+32) {
		tmp = t_0;
	} else if (x <= -1.95e-94) {
		tmp = t_1;
	} else if (x <= 1.35e-10) {
		tmp = fabs((4.0 / y));
	} else if (x <= 3.3e+107) {
		tmp = fabs((z * (x / y)));
	} else if ((x <= 3.4e+127) || !(x <= 6.2e+178)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((x / (y / z)))
    if (x <= (-1.35d+32)) then
        tmp = t_0
    else if (x <= (-1.95d-94)) then
        tmp = t_1
    else if (x <= 1.35d-10) then
        tmp = abs((4.0d0 / y))
    else if (x <= 3.3d+107) then
        tmp = abs((z * (x / y)))
    else if ((x <= 3.4d+127) .or. (.not. (x <= 6.2d+178))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((x / (y / z)));
	double tmp;
	if (x <= -1.35e+32) {
		tmp = t_0;
	} else if (x <= -1.95e-94) {
		tmp = t_1;
	} else if (x <= 1.35e-10) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 3.3e+107) {
		tmp = Math.abs((z * (x / y)));
	} else if ((x <= 3.4e+127) || !(x <= 6.2e+178)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x / (y / z)))
	tmp = 0
	if x <= -1.35e+32:
		tmp = t_0
	elif x <= -1.95e-94:
		tmp = t_1
	elif x <= 1.35e-10:
		tmp = math.fabs((4.0 / y))
	elif x <= 3.3e+107:
		tmp = math.fabs((z * (x / y)))
	elif (x <= 3.4e+127) or not (x <= 6.2e+178):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x / Float64(y / z)))
	tmp = 0.0
	if (x <= -1.35e+32)
		tmp = t_0;
	elseif (x <= -1.95e-94)
		tmp = t_1;
	elseif (x <= 1.35e-10)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 3.3e+107)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif ((x <= 3.4e+127) || !(x <= 6.2e+178))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x / (y / z)));
	tmp = 0.0;
	if (x <= -1.35e+32)
		tmp = t_0;
	elseif (x <= -1.95e-94)
		tmp = t_1;
	elseif (x <= 1.35e-10)
		tmp = abs((4.0 / y));
	elseif (x <= 3.3e+107)
		tmp = abs((z * (x / y)));
	elseif ((x <= 3.4e+127) || ~((x <= 6.2e+178)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.35e+32], t$95$0, If[LessEqual[x, -1.95e-94], t$95$1, If[LessEqual[x, 1.35e-10], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.3e+107], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 3.4e+127], N[Not[LessEqual[x, 6.2e+178]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35000000000000006e32 or 3.30000000000000032e107 < x < 3.39999999999999977e127 or 6.19999999999999982e178 < x

   1. Initial program 91.6%

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

      1. associate-*l/ 88.9%

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

      2. associate-*r/ 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in z around 0 74.0%

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

      1. associate-*r/ 74.0%

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

      2. metadata-eval 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in x around inf 74.0%

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

   if -1.35000000000000006e32 < x < -1.9500000000000001e-94 or 3.39999999999999977e127 < x < 6.19999999999999982e178

   1. Initial program 91.0%

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

      1. associate-*l/ 95.0%

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

      2. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in z around inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. associate-*r/ 67.4%

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

      3. *-commutative 67.4%

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

      4. distribute-rgt-neg-in 67.4%

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

      5. associate-*l/ 65.1%

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

   6. Simplified 65.1%

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

      1. expm1-log1p-u 48.5%

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

      2. expm1-udef 22.9%

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

      3. associate-/l* 22.9%

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

      4. add-sqr-sqrt 5.4%

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

      5. sqrt-unprod 7.9%

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

      6. sqr-neg 7.9%

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

      7. sqrt-unprod 4.5%

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

      8. add-sqr-sqrt 20.7%

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

   8. Applied egg-rr 20.7%

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

      1. expm1-def 36.8%

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

      2. expm1-log1p 57.9%

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

      3. associate-/l* 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-/l* 67.3%

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

   10. Simplified 67.3%

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

   if -1.9500000000000001e-94 < x < 1.35e-10

   1. Initial program 93.5%

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

      1. associate-*l/ 99.9%

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

      2. associate-*r/ 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in x around 0 82.4%

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

   if 1.35e-10 < x < 3.30000000000000032e107

   1. Initial program 94.8%

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

      1. associate-*l/ 90.1%

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

      2. associate-*r/ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in z around inf 56.9%

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

      1. mul-1-neg 56.9%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. distribute-rgt-neg-in 61.4%

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

      5. associate-*l/ 56.9%

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

   6. Simplified 56.9%

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

      1. expm1-log1p-u 23.3%

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

      2. expm1-udef 16.0%

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

      3. associate-/l* 16.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 37.7%

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

      6. sqr-neg 37.7%

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

      7. sqrt-unprod 37.7%

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

      8. add-sqr-sqrt 37.7%

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

   8. Applied egg-rr 37.7%

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

      1. expm1-def 45.0%

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

      2. expm1-log1p 61.5%

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

      3. associate-/l* 56.9%

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

      4. associate-*r/ 61.5%

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

   10. Simplified 61.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-94}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+107}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+127} \lor \neg \left(x \leq 6.2 \cdot 10^{+178}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 3: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+107} \lor \neg \left(x \leq 2.15 \cdot 10^{+130}\right) \land x \leq 5.2 \cdot 10^{+188}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -1.5)
     t_0
     (if (<= x 1.4e-10)
       (fabs (/ 4.0 y))
       (if (or (<= x 4.6e+107) (and (not (<= x 2.15e+130)) (<= x 5.2e+188)))
         (fabs (* z (/ x y)))
         t_0)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.4e-10) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 4.6e+107) || (!(x <= 2.15e+130) && (x <= 5.2e+188))) {
		tmp = fabs((z * (x / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if (x <= (-1.5d0)) then
        tmp = t_0
    else if (x <= 1.4d-10) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 4.6d+107) .or. (.not. (x <= 2.15d+130)) .and. (x <= 5.2d+188)) then
        tmp = abs((z * (x / y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.4e-10) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 4.6e+107) || (!(x <= 2.15e+130) && (x <= 5.2e+188))) {
		tmp = Math.abs((z * (x / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -1.5:
		tmp = t_0
	elif x <= 1.4e-10:
		tmp = math.fabs((4.0 / y))
	elif (x <= 4.6e+107) or (not (x <= 2.15e+130) and (x <= 5.2e+188)):
		tmp = math.fabs((z * (x / y)))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.4e-10)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 4.6e+107) || (!(x <= 2.15e+130) && (x <= 5.2e+188)))
		tmp = abs(Float64(z * Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.4e-10)
		tmp = abs((4.0 / y));
	elseif ((x <= 4.6e+107) || (~((x <= 2.15e+130)) && (x <= 5.2e+188)))
		tmp = abs((z * (x / y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.4e-10], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 4.6e+107], And[N[Not[LessEqual[x, 2.15e+130]], $MachinePrecision], LessEqual[x, 5.2e+188]]], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5 or 4.6000000000000001e107 < x < 2.14999999999999992e130 or 5.19999999999999975e188 < x

   1. Initial program 91.7%

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

      1. associate-*l/ 90.1%

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

      2. associate-*r/ 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in z around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. metadata-eval 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in x around inf 72.2%

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

   if -1.5 < x < 1.40000000000000008e-10

   1. Initial program 92.8%

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

      1. associate-*l/ 99.9%

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

      2. associate-*r/ 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in x around 0 74.4%

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

   if 1.40000000000000008e-10 < x < 4.6000000000000001e107 or 2.14999999999999992e130 < x < 5.19999999999999975e188

   1. Initial program 93.5%

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

      1. associate-*l/ 84.9%

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

      2. associate-*r/ 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in z around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. associate-*r/ 72.7%

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

      3. *-commutative 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

      5. associate-*l/ 64.3%

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

   6. Simplified 64.3%

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

      1. expm1-log1p-u 39.8%

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

      2. expm1-udef 32.2%

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

      3. associate-/l* 34.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 26.8%

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

      6. sqr-neg 26.8%

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

      7. sqrt-unprod 29.4%

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

      8. add-sqr-sqrt 29.4%

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

   8. Applied egg-rr 29.4%

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

      1. expm1-def 37.0%

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

      2. expm1-log1p 72.9%

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

      3. associate-/l* 64.3%

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

      4. associate-*r/ 72.9%

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

   10. Simplified 72.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+107} \lor \neg \left(x \leq 2.15 \cdot 10^{+130}\right) \land x \leq 5.2 \cdot 10^{+188}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+59}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -2.5e+59)
   (fabs (/ x (/ y z)))
   (if (<= z 6.5e-15)
     (fabs (/ (+ x 4.0) y))
     (fabs (* x (+ (/ z y) (/ -1.0 y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e+59) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 6.5e-15) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((x * ((z / y) + (-1.0 / y))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.5d+59)) then
        tmp = abs((x / (y / z)))
    else if (z <= 6.5d-15) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((x * ((z / y) + ((-1.0d0) / y))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e+59) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 6.5e-15) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((x * ((z / y) + (-1.0 / y))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -2.5e+59:
		tmp = math.fabs((x / (y / z)))
	elif z <= 6.5e-15:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((x * ((z / y) + (-1.0 / y))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -2.5e+59)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 6.5e-15)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(x * Float64(Float64(z / y) + Float64(-1.0 / y))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.5e+59)
		tmp = abs((x / (y / z)));
	elseif (z <= 6.5e-15)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((x * ((z / y) + (-1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -2.5e+59], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 6.5e-15], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(z / y), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4999999999999999e59

   1. Initial program 94.5%

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

      1. associate-*l/ 92.0%

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

      2. associate-*r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in z around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. associate-*r/ 76.7%

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

      3. *-commutative 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. associate-*l/ 74.8%

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

   6. Simplified 74.8%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-udef 26.0%

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

      3. associate-/l* 29.5%

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

      4. add-sqr-sqrt 14.2%

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

      5. sqrt-unprod 23.8%

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

      6. sqr-neg 23.8%

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

      7. sqrt-unprod 16.9%

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

      8. add-sqr-sqrt 32.8%

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

   8. Applied egg-rr 32.8%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 76.4%

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

      3. associate-/l* 74.8%

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

      4. *-commutative 74.8%

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

      5. associate-/l* 78.1%

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

   10. Simplified 78.1%

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

   if -2.4999999999999999e59 < z < 6.49999999999999991e-15

   1. Initial program 95.9%

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

      1. associate-*l/ 97.9%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in z around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate-*r/ 97.9%

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

      4. distribute-rgt-in 97.9%

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

      5. associate-*l/ 98.1%

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

      6. *-lft-identity 98.1%

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

   6. Simplified 98.1%

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

   if 6.49999999999999991e-15 < z

   1. Initial program 82.7%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in x around inf 81.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+59}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e-16)
   (fabs (/ (- (* x z) (+ x 4.0)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e-16) {
		tmp = fabs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.2d-16) then
        tmp = abs((((x * z) - (x + 4.0d0)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x * (z / y))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e-16) {
		tmp = Math.abs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 4.2e-16:
		tmp = math.fabs((((x * z) - (x + 4.0)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e-16)
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x * Float64(z / y))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.2e-16)
		tmp = abs((((x * z) - (x + 4.0)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 4.2e-16], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000002e-16

   1. Initial program 92.3%

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

      1. associate-*l/ 95.0%

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

      2. associate-*r/ 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around 0 96.3%

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

      1. sub-neg 96.3%

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

      2. +-commutative 96.3%

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

      3. distribute-lft-in 92.4%

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

      4. associate-+r+ 92.4%

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

      5. distribute-rgt-in 92.4%

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

      6. associate-*l/ 92.5%

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

      7. *-lft-identity 92.5%

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

      8. +-commutative 92.5%

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

      9. distribute-rgt-neg-out 92.5%

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

      10. sub-neg 92.5%

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

      11. associate-*r/ 95.0%

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

      12. div-sub 98.9%

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

   6. Simplified 98.9%

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

   if 4.2000000000000002e-16 < y

   1. Initial program 93.0%

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

      1. associate-*l/ 92.4%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+58)
   (fabs (/ x (/ y z)))
   (if (<= z 6.5e-15) (fabs (/ (+ x 4.0) y)) (fabs (/ x (/ y (- 1.0 z)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+58) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 6.5e-15) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.2d+58)) then
        tmp = abs((x / (y / z)))
    else if (z <= 6.5d-15) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((x / (y / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+58) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 6.5e-15) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -1.2e+58:
		tmp = math.fabs((x / (y / z)))
	elif z <= 6.5e-15:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((x / (y / (1.0 - z))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+58)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 6.5e-15)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.2e+58)
		tmp = abs((x / (y / z)));
	elseif (z <= 6.5e-15)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((x / (y / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -1.2e+58], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 6.5e-15], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2e58

   1. Initial program 94.5%

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

      1. associate-*l/ 92.0%

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

      2. associate-*r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in z around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. associate-*r/ 76.7%

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

      3. *-commutative 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. associate-*l/ 74.8%

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

   6. Simplified 74.8%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-udef 26.0%

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

      3. associate-/l* 29.5%

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

      4. add-sqr-sqrt 14.2%

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

      5. sqrt-unprod 23.8%

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

      6. sqr-neg 23.8%

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

      7. sqrt-unprod 16.9%

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

      8. add-sqr-sqrt 32.8%

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

   8. Applied egg-rr 32.8%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 76.4%

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

      3. associate-/l* 74.8%

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

      4. *-commutative 74.8%

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

      5. associate-/l* 78.1%

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

   10. Simplified 78.1%

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

   if -1.2e58 < z < 6.49999999999999991e-15

   1. Initial program 95.9%

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

      1. associate-*l/ 97.9%

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

      2. associate-*r/ 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in z around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate-*r/ 97.9%

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

      4. distribute-rgt-in 97.9%

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

      5. associate-*l/ 98.1%

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

      6. *-lft-identity 98.1%

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

   6. Simplified 98.1%

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

   if 6.49999999999999991e-15 < z

   1. Initial program 82.7%

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

      1. associate-*l/ 87.2%

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

      2. associate-*r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in x around 0 96.8%

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

      1. sub-neg 96.8%

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

      2. +-commutative 96.8%

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

      3. distribute-lft-in 90.2%

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

      4. associate-+r+ 90.2%

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

      5. distribute-rgt-in 90.2%

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

      6. associate-*l/ 90.2%

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

      7. *-lft-identity 90.2%

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

      8. +-commutative 90.2%

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

      9. distribute-rgt-neg-out 90.2%

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

      10. sub-neg 90.2%

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

      11. associate-*r/ 87.2%

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

      12. div-sub 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around inf 77.4%

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

      1. associate-/l* 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+58}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 7: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+95}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4e+95)
   (fabs (/ (- (* x z) (+ x 4.0)) y))
   (fabs (/ x (/ y (- 1.0 z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e+95) {
		tmp = fabs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = fabs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d+95) then
        tmp = abs((((x * z) - (x + 4.0d0)) / y))
    else
        tmp = abs((x / (y / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e+95) {
		tmp = Math.abs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = Math.abs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= 1.4e+95:
		tmp = math.fabs((((x * z) - (x + 4.0)) / y))
	else:
		tmp = math.fabs((x / (y / (1.0 - z))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4e+95)
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y));
	else
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.4e+95)
		tmp = abs((((x * z) - (x + 4.0)) / y));
	else
		tmp = abs((x / (y / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1.4e+95], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999e95

   1. Initial program 92.0%

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

      1. associate-*l/ 96.7%

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

      2. associate-*r/ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in x around 0 96.8%

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

      1. sub-neg 96.8%

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

      2. +-commutative 96.8%

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

      3. distribute-lft-in 94.5%

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

      4. associate-+r+ 94.5%

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

      5. distribute-rgt-in 94.4%

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

      6. associate-*l/ 94.5%

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

      7. *-lft-identity 94.5%

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

      8. +-commutative 94.5%

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

      9. distribute-rgt-neg-out 94.5%

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

      10. sub-neg 94.5%

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

      11. associate-*r/ 96.7%

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

      12. div-sub 99.0%

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

   6. Simplified 99.0%

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

   if 1.3999999999999999e95 < x

   1. Initial program 95.0%

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

      1. associate-*l/ 81.9%

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

      2. associate-*r/ 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-lft-in 94.8%

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

      4. associate-+r+ 94.8%

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

      5. distribute-rgt-in 94.8%

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

      6. associate-*l/ 95.1%

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

      7. *-lft-identity 95.1%

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

      8. +-commutative 95.1%

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

      9. distribute-rgt-neg-out 95.1%

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

      10. sub-neg 95.1%

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

      11. associate-*r/ 81.9%

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

      12. div-sub 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in x around inf 86.6%

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

      1. associate-/l* 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+95}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 8: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+57} \lor \neg \left(z \leq 10^{+21}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.6e+57) (not (<= z 1e+21)))
   (fabs (/ x (/ y z)))
   (fabs (/ (+ x 4.0) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.6e+57) || !(z <= 1e+21)) {
		tmp = fabs((x / (y / z)));
	} else {
		tmp = fabs(((x + 4.0) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.6d+57)) .or. (.not. (z <= 1d+21))) then
        tmp = abs((x / (y / z)))
    else
        tmp = abs(((x + 4.0d0) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.6e+57) || !(z <= 1e+21)) {
		tmp = Math.abs((x / (y / z)));
	} else {
		tmp = Math.abs(((x + 4.0) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (z <= -6.6e+57) or not (z <= 1e+21):
		tmp = math.fabs((x / (y / z)))
	else:
		tmp = math.fabs(((x + 4.0) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.6e+57) || !(z <= 1e+21))
		tmp = abs(Float64(x / Float64(y / z)));
	else
		tmp = abs(Float64(Float64(x + 4.0) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.6e+57) || ~((z <= 1e+21)))
		tmp = abs((x / (y / z)));
	else
		tmp = abs(((x + 4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[z, -6.6e+57], N[Not[LessEqual[z, 1e+21]], $MachinePrecision]], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6000000000000002e57 or 1e21 < z

   1. Initial program 86.6%

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

      1. associate-*l/ 89.2%

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

      2. associate-*r/ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in z around inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. associate-*r/ 80.2%

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

      3. *-commutative 80.2%

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

      4. distribute-rgt-neg-in 80.2%

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

      5. associate-*l/ 77.5%

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

   6. Simplified 77.5%

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

      1. expm1-log1p-u 47.3%

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

      2. expm1-udef 26.4%

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

      3. associate-/l* 28.9%

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

      4. add-sqr-sqrt 14.4%

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

      5. sqrt-unprod 21.7%

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

      6. sqr-neg 21.7%

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

      7. sqrt-unprod 15.2%

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

      8. add-sqr-sqrt 34.2%

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

   8. Applied egg-rr 34.2%

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

      1. expm1-def 45.1%

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

      2. expm1-log1p 74.2%

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

      3. associate-/l* 77.5%

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

      4. *-commutative 77.5%

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

      5. associate-/l* 80.9%

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

   10. Simplified 80.9%

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

   if -6.6000000000000002e57 < z < 1e21

   1. Initial program 96.2%

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

      1. associate-*l/ 97.4%

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

      2. associate-*r/ 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in z around 0 95.0%

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

      1. +-commutative 95.0%

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

      2. *-rgt-identity 95.0%

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

      3. associate-*r/ 94.9%

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

      4. distribute-rgt-in 94.8%

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

      5. associate-*l/ 95.0%

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

      6. *-lft-identity 95.0%

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

   6. Simplified 95.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+57} \lor \neg \left(z \leq 10^{+21}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 9: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -5.8e+57)
   (fabs (/ x (/ y z)))
   (if (<= z 9.5e+20) (fabs (/ (+ x 4.0) y)) (fabs (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+57) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 9.5e+20) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((x * (z / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.8d+57)) then
        tmp = abs((x / (y / z)))
    else if (z <= 9.5d+20) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+57) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 9.5e+20) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -5.8e+57:
		tmp = math.fabs((x / (y / z)))
	elif z <= 9.5e+20:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -5.8e+57)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 9.5e+20)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.8e+57)
		tmp = abs((x / (y / z)));
	elseif (z <= 9.5e+20)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -5.8e+57], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 9.5e+20], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.8000000000000003e57

   1. Initial program 94.5%

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

      1. associate-*l/ 92.0%

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

      2. associate-*r/ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in z around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. associate-*r/ 76.7%

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

      3. *-commutative 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. associate-*l/ 74.8%

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

   6. Simplified 74.8%

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

      1. expm1-log1p-u 46.7%

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

      2. expm1-udef 26.0%

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

      3. associate-/l* 29.5%

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

      4. add-sqr-sqrt 14.2%

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

      5. sqrt-unprod 23.8%

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

      6. sqr-neg 23.8%

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

      7. sqrt-unprod 16.9%

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

      8. add-sqr-sqrt 32.8%

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

   8. Applied egg-rr 32.8%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 76.4%

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

      3. associate-/l* 74.8%

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

      4. *-commutative 74.8%

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

      5. associate-/l* 78.1%

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

   10. Simplified 78.1%

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

   if -5.8000000000000003e57 < z < 9.5e20

   1. Initial program 96.2%

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

      1. associate-*l/ 97.4%

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

      2. associate-*r/ 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in z around 0 95.0%

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

      1. +-commutative 95.0%

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

      2. *-rgt-identity 95.0%

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

      3. associate-*r/ 94.9%

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

      4. distribute-rgt-in 94.8%

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

      5. associate-*l/ 95.0%

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

      6. *-lft-identity 95.0%

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

   6. Simplified 95.0%

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

   if 9.5e20 < z

   1. Initial program 79.0%

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

      1. associate-*l/ 86.4%

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

      2. associate-*r/ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. +-commutative 98.0%

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

      3. distribute-lft-in 90.0%

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

      4. associate-+r+ 90.0%

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

      5. distribute-rgt-in 90.0%

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

      6. associate-*l/ 90.0%

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

      7. *-lft-identity 90.0%

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

      8. +-commutative 90.0%

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

      9. distribute-rgt-neg-out 90.0%

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

      10. sub-neg 90.0%

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

      11. associate-*r/ 86.4%

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

      12. div-sub 94.4%

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

   6. Simplified 94.4%

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

   7. Taylor expanded in z around inf 80.1%

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

      1. associate-*r/ 80.1%

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

      2. mul-1-neg 80.1%

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

      3. distribute-rgt-neg-out 80.1%

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

      4. associate-*r/ 83.6%

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

   9. Simplified 83.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 10: 68.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 4 < x

   1. Initial program 92.1%

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

      1. associate-*l/ 88.7%

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

      2. associate-*r/ 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in z around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. metadata-eval 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in x around inf 64.0%

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

   if -1.5 < x < 4

   1. Initial program 92.9%

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

      1. associate-*l/ 99.9%

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

      2. associate-*r/ 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around 0 73.4%

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

4. Final simplification 68.7%

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

Alternative 11: 40.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

   1. associate-*l/ 94.3%

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

   2. associate-*r/ 94.6%

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

3. Simplified 94.6%

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

4. Taylor expanded in x around 0 39.2%

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

5. Final simplification 39.2%

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

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))