fabs fraction 1

Percentage Accurate: 91.8% → 99.9%

Time: 7.3s

Alternatives: 10

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-6)
   (fabs (/ (- (+ x 4.0) (* x z)) 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-6) {
		tmp = fabs((((x + 4.0) - (x * z)) / 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-6)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / 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-6], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $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 < 3.99999999999999982e-6

   1. Initial program 90.9%

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

      1. associate-*l/ 93.0%

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

      2. sub-div 97.3%

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

   3. Applied egg-rr 97.3%

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

   if 3.99999999999999982e-6 < y

   1. Initial program 97.6%

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

      1. fabs-sub 97.6%

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

      2. associate-*l/ 91.9%

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

      3. *-commutative 91.9%

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

      4. associate-*l/ 99.9%

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

      5. *-commutative 99.9%

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

      6. fma-neg 99.9%

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

      7. distribute-neg-frac 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. 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%

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

4. Final simplification 98.1%

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

Alternative 2: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -1.35e+121)
     t_0
     (if (<= x -5.2e+43)
       t_1
       (if (<= x -2.25e-73)
         t_0
         (if (<= x 5e-56) (fabs (/ 4.0 y)) (if (<= x 1.32e+44) t_0 t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.35e+121) {
		tmp = t_0;
	} else if (x <= -5.2e+43) {
		tmp = t_1;
	} else if (x <= -2.25e-73) {
		tmp = t_0;
	} else if (x <= 5e-56) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.32e+44) {
		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((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-1.35d+121)) then
        tmp = t_0
    else if (x <= (-5.2d+43)) then
        tmp = t_1
    else if (x <= (-2.25d-73)) then
        tmp = t_0
    else if (x <= 5d-56) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.32d+44) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.35e+121) {
		tmp = t_0;
	} else if (x <= -5.2e+43) {
		tmp = t_1;
	} else if (x <= -2.25e-73) {
		tmp = t_0;
	} else if (x <= 5e-56) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.35e+121:
		tmp = t_0
	elif x <= -5.2e+43:
		tmp = t_1
	elif x <= -2.25e-73:
		tmp = t_0
	elif x <= 5e-56:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.32e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.35e+121)
		tmp = t_0;
	elseif (x <= -5.2e+43)
		tmp = t_1;
	elseif (x <= -2.25e-73)
		tmp = t_0;
	elseif (x <= 5e-56)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.35e+121)
		tmp = t_0;
	elseif (x <= -5.2e+43)
		tmp = t_1;
	elseif (x <= -2.25e-73)
		tmp = t_0;
	elseif (x <= 5e-56)
		tmp = abs((4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.35e+121], t$95$0, If[LessEqual[x, -5.2e+43], t$95$1, If[LessEqual[x, -2.25e-73], t$95$0, If[LessEqual[x, 5e-56], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.32e+44], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e121 or -5.20000000000000042e43 < x < -2.25e-73 or 4.99999999999999997e-56 < x < 1.3200000000000001e44

   1. Initial program 90.2%

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

   2. Taylor expanded in z around inf 61.7%

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

      1. associate-*r/ 61.7%

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

      2. associate-*r* 61.7%

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

      3. neg-mul-1 61.7%

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

      4. associate-*r/ 69.8%

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

      5. *-commutative 69.8%

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

   4. Simplified 69.8%

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

      1. expm1-log1p-u 38.3%

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

      2. expm1-udef 30.2%

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

      3. add-sqr-sqrt 11.7%

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

      4. sqrt-unprod 19.9%

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

      5. sqr-neg 19.9%

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

      6. sqrt-unprod 13.0%

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

      7. add-sqr-sqrt 30.1%

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

      8. associate-*l/ 25.5%

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

      9. associate-/l* 27.7%

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

   6. Applied egg-rr 27.7%

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

      1. expm1-def 36.4%

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

      2. expm1-log1p 64.6%

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

      3. associate-/r/ 69.8%

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

      4. *-commutative 69.8%

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

   8. Simplified 69.8%

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

   if -1.3500000000000001e121 < x < -5.20000000000000042e43 or 1.3200000000000001e44 < x

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-rgt-in 96.0%

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

      5. associate-*l/ 96.1%

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

      6. *-lft-identity 96.1%

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

      7. associate-*r* 96.1%

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

      8. associate-*l/ 87.7%

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

      9. associate-*r/ 87.7%

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

      10. associate-*r* 87.7%

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

      11. neg-mul-1 87.7%

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

      12. associate-*r/ 89.7%

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

      13. distribute-rgt1-in 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.7%

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

   if -2.25e-73 < x < 4.99999999999999997e-56

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 81.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+121}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -4.3e+120)
     t_0
     (if (<= x -3.4e+42)
       t_1
       (if (<= x -1.35e-73)
         t_0
         (if (<= x 2.85e-55)
           (fabs (/ 4.0 y))
           (if (<= x 1.32e+44) (fabs (* x (/ z y))) t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -4.3e+120) {
		tmp = t_0;
	} else if (x <= -3.4e+42) {
		tmp = t_1;
	} else if (x <= -1.35e-73) {
		tmp = t_0;
	} else if (x <= 2.85e-55) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = fabs((x * (z / y)));
	} 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((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-4.3d+120)) then
        tmp = t_0
    else if (x <= (-3.4d+42)) then
        tmp = t_1
    else if (x <= (-1.35d-73)) then
        tmp = t_0
    else if (x <= 2.85d-55) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.32d+44) then
        tmp = abs((x * (z / y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -4.3e+120) {
		tmp = t_0;
	} else if (x <= -3.4e+42) {
		tmp = t_1;
	} else if (x <= -1.35e-73) {
		tmp = t_0;
	} else if (x <= 2.85e-55) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -4.3e+120:
		tmp = t_0
	elif x <= -3.4e+42:
		tmp = t_1
	elif x <= -1.35e-73:
		tmp = t_0
	elif x <= 2.85e-55:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.32e+44:
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -4.3e+120)
		tmp = t_0;
	elseif (x <= -3.4e+42)
		tmp = t_1;
	elseif (x <= -1.35e-73)
		tmp = t_0;
	elseif (x <= 2.85e-55)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -4.3e+120)
		tmp = t_0;
	elseif (x <= -3.4e+42)
		tmp = t_1;
	elseif (x <= -1.35e-73)
		tmp = t_0;
	elseif (x <= 2.85e-55)
		tmp = abs((4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = abs((x * (z / y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.3e+120], t$95$0, If[LessEqual[x, -3.4e+42], t$95$1, If[LessEqual[x, -1.35e-73], t$95$0, If[LessEqual[x, 2.85e-55], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.32e+44], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.3000000000000002e120 or -3.39999999999999975e42 < x < -1.34999999999999997e-73

   1. Initial program 86.1%

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

   2. Taylor expanded in z around inf 59.1%

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

      1. associate-*r/ 59.1%

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

      2. associate-*r* 59.1%

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

      3. neg-mul-1 59.1%

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

      4. associate-*r/ 72.0%

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

      5. *-commutative 72.0%

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

   4. Simplified 72.0%

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

      1. expm1-log1p-u 39.0%

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

      2. expm1-udef 35.2%

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

      3. add-sqr-sqrt 9.7%

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

      4. sqrt-unprod 21.4%

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

      5. sqr-neg 21.4%

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

      6. sqrt-unprod 13.6%

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

      7. add-sqr-sqrt 31.1%

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

      8. associate-*l/ 24.2%

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

      9. associate-/l* 27.6%

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

   6. Applied egg-rr 27.6%

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

      1. expm1-def 31.4%

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

      2. expm1-log1p 63.2%

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

      3. associate-/r/ 72.0%

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

      4. *-commutative 72.0%

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

   8. Simplified 72.0%

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

   if -4.3000000000000002e120 < x < -3.39999999999999975e42 or 1.3200000000000001e44 < x

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-rgt-in 96.0%

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

      5. associate-*l/ 96.1%

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

      6. *-lft-identity 96.1%

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

      7. associate-*r* 96.1%

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

      8. associate-*l/ 87.7%

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

      9. associate-*r/ 87.7%

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

      10. associate-*r* 87.7%

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

      11. neg-mul-1 87.7%

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

      12. associate-*r/ 89.7%

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

      13. distribute-rgt1-in 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.7%

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

   if -1.34999999999999997e-73 < x < 2.8500000000000001e-55

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 81.8%

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

   if 2.8500000000000001e-55 < x < 1.3200000000000001e44

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. associate-*r* 66.9%

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

      3. neg-mul-1 66.9%

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

      4. associate-*r/ 65.5%

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

      5. *-commutative 65.5%

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

   4. Simplified 65.5%

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

      1. expm1-log1p-u 37.0%

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

      2. expm1-udef 20.4%

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

      3. add-sqr-sqrt 15.6%

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

      4. sqrt-unprod 17.0%

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

      5. sqr-neg 17.0%

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

      6. sqrt-unprod 11.7%

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

      7. add-sqr-sqrt 28.1%

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

      8. associate-*l/ 28.1%

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

      9. associate-/l* 28.1%

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

   6. Applied egg-rr 28.1%

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

      1. expm1-def 46.3%

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

      2. expm1-log1p 67.1%

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

   8. Simplified 67.1%

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

      1. clear-num 67.0%

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

      2. associate-/r/ 67.1%

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

      3. clear-num 67.1%

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

   10. Applied egg-rr 67.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+120}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-73}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -3.2e+120)
     t_0
     (if (<= x -1.65e+41)
       t_1
       (if (<= x -2.25e-73)
         t_0
         (if (<= x 2.85e-55)
           (fabs (/ 4.0 y))
           (if (<= x 1.32e+44) (fabs (/ x (/ y z))) t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -3.2e+120) {
		tmp = t_0;
	} else if (x <= -1.65e+41) {
		tmp = t_1;
	} else if (x <= -2.25e-73) {
		tmp = t_0;
	} else if (x <= 2.85e-55) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = fabs((x / (y / z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-3.2d+120)) then
        tmp = t_0
    else if (x <= (-1.65d+41)) then
        tmp = t_1
    else if (x <= (-2.25d-73)) then
        tmp = t_0
    else if (x <= 2.85d-55) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.32d+44) then
        tmp = abs((x / (y / z)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -3.2e+120) {
		tmp = t_0;
	} else if (x <= -1.65e+41) {
		tmp = t_1;
	} else if (x <= -2.25e-73) {
		tmp = t_0;
	} else if (x <= 2.85e-55) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = Math.abs((x / (y / z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -3.2e+120:
		tmp = t_0
	elif x <= -1.65e+41:
		tmp = t_1
	elif x <= -2.25e-73:
		tmp = t_0
	elif x <= 2.85e-55:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.32e+44:
		tmp = math.fabs((x / (y / z)))
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -3.2e+120)
		tmp = t_0;
	elseif (x <= -1.65e+41)
		tmp = t_1;
	elseif (x <= -2.25e-73)
		tmp = t_0;
	elseif (x <= 2.85e-55)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = abs(Float64(x / Float64(y / z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -3.2e+120)
		tmp = t_0;
	elseif (x <= -1.65e+41)
		tmp = t_1;
	elseif (x <= -2.25e-73)
		tmp = t_0;
	elseif (x <= 2.85e-55)
		tmp = abs((4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = abs((x / (y / z)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.2e+120], t$95$0, If[LessEqual[x, -1.65e+41], t$95$1, If[LessEqual[x, -2.25e-73], t$95$0, If[LessEqual[x, 2.85e-55], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.32e+44], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.19999999999999982e120 or -1.65e41 < x < -2.25e-73

   1. Initial program 86.1%

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

   2. Taylor expanded in z around inf 59.1%

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

      1. associate-*r/ 59.1%

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

      2. associate-*r* 59.1%

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

      3. neg-mul-1 59.1%

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

      4. associate-*r/ 72.0%

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

      5. *-commutative 72.0%

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

   4. Simplified 72.0%

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

      1. expm1-log1p-u 39.0%

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

      2. expm1-udef 35.2%

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

      3. add-sqr-sqrt 9.7%

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

      4. sqrt-unprod 21.4%

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

      5. sqr-neg 21.4%

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

      6. sqrt-unprod 13.6%

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

      7. add-sqr-sqrt 31.1%

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

      8. associate-*l/ 24.2%

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

      9. associate-/l* 27.6%

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

   6. Applied egg-rr 27.6%

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

      1. expm1-def 31.4%

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

      2. expm1-log1p 63.2%

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

      3. associate-/r/ 72.0%

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

      4. *-commutative 72.0%

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

   8. Simplified 72.0%

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

   if -3.19999999999999982e120 < x < -1.65e41 or 1.3200000000000001e44 < x

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-rgt-in 96.0%

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

      5. associate-*l/ 96.1%

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

      6. *-lft-identity 96.1%

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

      7. associate-*r* 96.1%

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

      8. associate-*l/ 87.7%

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

      9. associate-*r/ 87.7%

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

      10. associate-*r* 87.7%

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

      11. neg-mul-1 87.7%

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

      12. associate-*r/ 89.7%

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

      13. distribute-rgt1-in 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.7%

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

   if -2.25e-73 < x < 2.8500000000000001e-55

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 81.8%

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

   if 2.8500000000000001e-55 < x < 1.3200000000000001e44

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. associate-*r* 66.9%

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

      3. neg-mul-1 66.9%

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

      4. associate-*r/ 65.5%

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

      5. *-commutative 65.5%

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

   4. Simplified 65.5%

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

      1. expm1-log1p-u 37.0%

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

      2. expm1-udef 20.4%

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

      3. add-sqr-sqrt 15.6%

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

      4. sqrt-unprod 17.0%

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

      5. sqr-neg 17.0%

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

      6. sqrt-unprod 11.7%

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

      7. add-sqr-sqrt 28.1%

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

      8. associate-*l/ 28.1%

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

      9. associate-/l* 28.1%

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

   6. Applied egg-rr 28.1%

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

      1. expm1-def 46.3%

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

      2. expm1-log1p 67.1%

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

   8. Simplified 67.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+120}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+120}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -1.3e+120)
     (fabs (* z (/ x y)))
     (if (<= x -2.8e+42)
       t_0
       (if (<= x -2.05e-73)
         (fabs (/ (* x z) y))
         (if (<= x 2.85e-55)
           (fabs (/ 4.0 y))
           (if (<= x 1.32e+44) (fabs (/ x (/ y z))) t_0)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -1.3e+120) {
		tmp = fabs((z * (x / y)));
	} else if (x <= -2.8e+42) {
		tmp = t_0;
	} else if (x <= -2.05e-73) {
		tmp = fabs(((x * z) / y));
	} else if (x <= 2.85e-55) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = fabs((x / (y / z)));
	} 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 <= (-1.3d+120)) then
        tmp = abs((z * (x / y)))
    else if (x <= (-2.8d+42)) then
        tmp = t_0
    else if (x <= (-2.05d-73)) then
        tmp = abs(((x * z) / y))
    else if (x <= 2.85d-55) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.32d+44) then
        tmp = abs((x / (y / z)))
    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 <= -1.3e+120) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= -2.8e+42) {
		tmp = t_0;
	} else if (x <= -2.05e-73) {
		tmp = Math.abs(((x * z) / y));
	} else if (x <= 2.85e-55) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.32e+44) {
		tmp = Math.abs((x / (y / z)));
	} 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 <= -1.3e+120:
		tmp = math.fabs((z * (x / y)))
	elif x <= -2.8e+42:
		tmp = t_0
	elif x <= -2.05e-73:
		tmp = math.fabs(((x * z) / y))
	elif x <= 2.85e-55:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.32e+44:
		tmp = math.fabs((x / (y / z)))
	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 <= -1.3e+120)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= -2.8e+42)
		tmp = t_0;
	elseif (x <= -2.05e-73)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (x <= 2.85e-55)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = abs(Float64(x / Float64(y / z)));
	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 <= -1.3e+120)
		tmp = abs((z * (x / y)));
	elseif (x <= -2.8e+42)
		tmp = t_0;
	elseif (x <= -2.05e-73)
		tmp = abs(((x * z) / y));
	elseif (x <= 2.85e-55)
		tmp = abs((4.0 / y));
	elseif (x <= 1.32e+44)
		tmp = abs((x / (y / z)));
	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, -1.3e+120], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -2.8e+42], t$95$0, If[LessEqual[x, -2.05e-73], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.85e-55], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.32e+44], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.2999999999999999e120

   1. Initial program 77.3%

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

   2. Taylor expanded in z around inf 57.1%

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

      1. associate-*r/ 57.1%

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

      2. associate-*r* 57.1%

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

      3. neg-mul-1 57.1%

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

      4. associate-*r/ 78.4%

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

      5. *-commutative 78.4%

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

   4. Simplified 78.4%

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

      1. expm1-log1p-u 44.7%

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

      2. expm1-udef 44.6%

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

      3. add-sqr-sqrt 15.2%

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

      4. sqrt-unprod 24.9%

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

      5. sqr-neg 24.9%

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

      6. sqrt-unprod 12.9%

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

      7. add-sqr-sqrt 31.6%

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

      8. associate-*l/ 20.3%

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

      9. associate-/l* 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 28.6%

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

      2. expm1-log1p 69.2%

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

      3. associate-/r/ 78.4%

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

      4. *-commutative 78.4%

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

   8. Simplified 78.4%

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

   if -1.2999999999999999e120 < x < -2.7999999999999999e42 or 1.3200000000000001e44 < x

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-rgt-in 96.0%

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

      5. associate-*l/ 96.1%

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

      6. *-lft-identity 96.1%

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

      7. associate-*r* 96.1%

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

      8. associate-*l/ 87.7%

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

      9. associate-*r/ 87.7%

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

      10. associate-*r* 87.7%

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

      11. neg-mul-1 87.7%

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

      12. associate-*r/ 89.7%

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

      13. distribute-rgt1-in 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.7%

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

   if -2.7999999999999999e42 < x < -2.05000000000000008e-73

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 62.2%

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

      1. associate-*r/ 62.2%

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

      2. associate-*r* 62.2%

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

      3. neg-mul-1 62.2%

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

      4. associate-*r/ 62.0%

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

      5. *-commutative 62.0%

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

   4. Simplified 62.0%

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

      1. add-sqr-sqrt 16.4%

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

      2. sqrt-unprod 52.7%

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

      3. sqr-neg 52.7%

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

      4. sqrt-unprod 45.2%

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

      5. add-sqr-sqrt 62.0%

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

      6. associate-*l/ 62.2%

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

   6. Applied egg-rr 62.2%

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

   if -2.05000000000000008e-73 < x < 2.8500000000000001e-55

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 81.8%

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

   if 2.8500000000000001e-55 < x < 1.3200000000000001e44

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. associate-*r* 66.9%

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

      3. neg-mul-1 66.9%

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

      4. associate-*r/ 65.5%

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

      5. *-commutative 65.5%

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

   4. Simplified 65.5%

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

      1. expm1-log1p-u 37.0%

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

      2. expm1-udef 20.4%

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

      3. add-sqr-sqrt 15.6%

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

      4. sqrt-unprod 17.0%

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

      5. sqr-neg 17.0%

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

      6. sqrt-unprod 11.7%

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

      7. add-sqr-sqrt 28.1%

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

      8. associate-*l/ 28.1%

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

      9. associate-/l* 28.1%

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

   6. Applied egg-rr 28.1%

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

      1. expm1-def 46.3%

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

      2. expm1-log1p 67.1%

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

   8. Simplified 67.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+120}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+42}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0004:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0004)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* z (/ x y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0004) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (z * (x / y))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.0004d0) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (z * (x / y))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0004) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (z * (x / y))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 0.0004:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (z * (x / y))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0004)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0004)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (z * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 0.0004], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000019e-4

   1. Initial program 90.9%

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

      1. associate-*l/ 93.0%

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

      2. sub-div 97.3%

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

   3. Applied egg-rr 97.3%

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

   if 4.00000000000000019e-4 < y

   1. Initial program 97.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0004:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+143)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (* (/ x y) (- 1.0 z)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+143) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x / y) * (1.0 - z)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2d+143) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x / y) * (1.0d0 - z)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+143) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= 2e+143:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+143)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2e+143)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x / y) * (1.0 - z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 2e+143], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e143

   1. Initial program 93.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. sub-div 98.2%

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

   3. Applied egg-rr 98.2%

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

   if 2e143 < x

   1. Initial program 91.1%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-rgt-in 96.9%

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

      5. associate-*l/ 97.0%

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

      6. *-lft-identity 97.0%

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

      7. associate-*r* 97.0%

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

      8. associate-*l/ 77.7%

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

      9. associate-*r/ 77.7%

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

      10. associate-*r* 77.7%

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

      11. neg-mul-1 77.7%

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

      12. associate-*r/ 91.1%

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

      13. distribute-rgt1-in 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]

Alternative 8: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e+18)
   (fabs (/ x (/ y z)))
   (if (<= z 2.3e-10) (fabs (/ (- -4.0 x) y)) (fabs (* z (/ x y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+18) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 2.3e-10) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z * (x / 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 <= (-2.2d+18)) then
        tmp = abs((x / (y / z)))
    else if (z <= 2.3d-10) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z * (x / 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 <= -2.2e+18) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 2.3e-10) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -2.2e+18:
		tmp = math.fabs((x / (y / z)))
	elif z <= 2.3e-10:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e+18)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 2.3e-10)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2e+18)
		tmp = abs((x / (y / z)));
	elseif (z <= 2.3e-10)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -2.2e+18], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.3e-10], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e18

   1. Initial program 96.2%

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

   2. Taylor expanded in z around inf 64.9%

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

      1. associate-*r/ 64.9%

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

      2. associate-*r* 64.9%

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

      3. neg-mul-1 64.9%

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

      4. associate-*r/ 73.0%

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

      5. *-commutative 73.0%

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

   4. Simplified 73.0%

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

      1. expm1-log1p-u 36.0%

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

      2. expm1-udef 24.4%

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

      3. add-sqr-sqrt 24.4%

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

      4. sqrt-unprod 17.5%

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

      5. sqr-neg 17.5%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 35.6%

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

      8. associate-*l/ 30.3%

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

      9. associate-/l* 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. expm1-def 49.7%

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

      2. expm1-log1p 75.0%

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

   8. Simplified 75.0%

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

   if -2.2e18 < z < 2.30000000000000007e-10

   1. Initial program 95.0%

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

      1. fabs-sub 95.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 98.9%

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

      1. associate-*r/ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. metadata-eval 98.9%

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

      4. neg-mul-1 98.9%

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

      5. sub-neg 98.9%

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

   6. Simplified 98.9%

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

   if 2.30000000000000007e-10 < z

   1. Initial program 83.0%

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

   2. Taylor expanded in z around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. associate-*r* 80.4%

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

      3. neg-mul-1 80.4%

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

      4. associate-*r/ 85.7%

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

      5. *-commutative 85.7%

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

   4. Simplified 85.7%

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

      1. expm1-log1p-u 53.7%

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

      2. expm1-udef 41.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 21.3%

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

      5. sqr-neg 21.3%

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

      6. sqrt-unprod 29.7%

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

      7. add-sqr-sqrt 29.7%

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

      8. associate-*l/ 24.8%

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

      9. associate-/l* 26.2%

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

   6. Applied egg-rr 26.2%

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

      1. expm1-def 40.5%

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

      2. expm1-log1p 82.1%

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

      3. associate-/r/ 85.7%

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

      4. *-commutative 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 9: 68.8% 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 88.1%

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

   2. Taylor expanded in x around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. distribute-rgt-in 90.4%

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

      5. associate-*l/ 90.5%

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

      6. *-lft-identity 90.5%

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

      7. associate-*r* 90.5%

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

      8. associate-*l/ 82.4%

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

      9. associate-*r/ 82.4%

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

      10. associate-*r* 82.4%

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

      11. neg-mul-1 82.4%

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

      12. associate-*r/ 85.0%

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

      13. distribute-rgt1-in 96.8%

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

      14. +-commutative 96.8%

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

      15. sub-neg 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in z around 0 63.8%

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

   if -1.5 < x < 4

   1. Initial program 97.4%

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

   2. Taylor expanded in x around 0 71.3%

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

4. Final simplification 67.6%

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

Alternative 10: 39.4% 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 92.8%

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

2. Taylor expanded in x around 0 38.6%

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

3. Final simplification 38.6%

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

Reproduce

herbie shell --seed 2023217 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))