fabs fraction 1

Percentage Accurate: 91.0% → 98.1%

Time: 7.9s

Alternatives: 12

Speedup: 1.0×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% 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: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(z + -1\right) \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.5e+16)
   (fabs (/ (- x (fma x z -4.0)) y))
   (fabs (* (+ z -1.0) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.5e+16) {
		tmp = fabs(((x - fma(x, z, -4.0)) / y));
	} else {
		tmp = fabs(((z + -1.0) * (x / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.5e+16)
		tmp = abs(Float64(Float64(x - fma(x, z, -4.0)) / y));
	else
		tmp = abs(Float64(Float64(z + -1.0) * Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.5e+16], N[Abs[N[(N[(x - N[(x * z + -4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e16

   1. Initial program 92.1%

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

      1. fabs-neg 92.1%

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

      2. sub-neg 92.1%

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

      3. distribute-neg-in 92.1%

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

      4. sub-neg 92.1%

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

      5. distribute-neg-frac 92.1%

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

      6. associate-*l/ 95.2%

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

      7. distribute-neg-frac 95.2%

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

      8. neg-mul-1 95.2%

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

      9. associate-*l/ 95.2%

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

      10. neg-mul-1 95.2%

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

      11. associate-*l/ 95.2%

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

      12. distribute-lft-out-- 98.9%

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

      13. fabs-mul 98.9%

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

      14. fabs-sub 98.9%

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

      15. fabs-mul 98.9%

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

      16. associate-*l/ 98.9%

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

   3. Simplified 98.9%

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

   if 5.5e16 < x

   1. Initial program 86.2%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around inf 89.8%

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

      1. associate-/l* 99.8%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(z + -1\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 2: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+81}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))) (t_2 (fabs (/ 4.0 y))))
   (if (<= x -10.5)
     t_1
     (if (<= x 6.5e-90)
       t_2
       (if (<= x 2.95e-37)
         t_0
         (if (<= x 4.0)
           t_2
           (if (<= x 1.95e+49)
             t_1
             (if (<= x 3e+81)
               (fabs (* x (/ z y)))
               (if (<= x 1.45e+162) t_1 t_0)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double t_2 = fabs((4.0 / y));
	double tmp;
	if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 6.5e-90) {
		tmp = t_2;
	} else if (x <= 2.95e-37) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = t_2;
	} else if (x <= 1.95e+49) {
		tmp = t_1;
	} else if (x <= 3e+81) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 1.45e+162) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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) :: t_2
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    t_2 = abs((4.0d0 / y))
    if (x <= (-10.5d0)) then
        tmp = t_1
    else if (x <= 6.5d-90) then
        tmp = t_2
    else if (x <= 2.95d-37) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = t_2
    else if (x <= 1.95d+49) then
        tmp = t_1
    else if (x <= 3d+81) then
        tmp = abs((x * (z / y)))
    else if (x <= 1.45d+162) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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 t_2 = Math.abs((4.0 / y));
	double tmp;
	if (x <= -10.5) {
		tmp = t_1;
	} else if (x <= 6.5e-90) {
		tmp = t_2;
	} else if (x <= 2.95e-37) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = t_2;
	} else if (x <= 1.95e+49) {
		tmp = t_1;
	} else if (x <= 3e+81) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 1.45e+162) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	t_2 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -10.5:
		tmp = t_1
	elif x <= 6.5e-90:
		tmp = t_2
	elif x <= 2.95e-37:
		tmp = t_0
	elif x <= 4.0:
		tmp = t_2
	elif x <= 1.95e+49:
		tmp = t_1
	elif x <= 3e+81:
		tmp = math.fabs((x * (z / y)))
	elif x <= 1.45e+162:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	t_2 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_1;
	elseif (x <= 6.5e-90)
		tmp = t_2;
	elseif (x <= 2.95e-37)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = t_2;
	elseif (x <= 1.95e+49)
		tmp = t_1;
	elseif (x <= 3e+81)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 1.45e+162)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	t_2 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_1;
	elseif (x <= 6.5e-90)
		tmp = t_2;
	elseif (x <= 2.95e-37)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = t_2;
	elseif (x <= 1.95e+49)
		tmp = t_1;
	elseif (x <= 3e+81)
		tmp = abs((x * (z / y)));
	elseif (x <= 1.45e+162)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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]}, Block[{t$95$2 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$1, If[LessEqual[x, 6.5e-90], t$95$2, If[LessEqual[x, 2.95e-37], t$95$0, If[LessEqual[x, 4.0], t$95$2, If[LessEqual[x, 1.95e+49], t$95$1, If[LessEqual[x, 3e+81], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.45e+162], t$95$1, t$95$0]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -10.5 or 4 < x < 1.95e49 or 2.99999999999999997e81 < x < 1.45000000000000003e162

   1. Initial program 84.1%

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

      1. associate-*l/ 85.6%

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

      2. sub-div 96.6%

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

   3. Applied egg-rr 96.6%

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

   4. Taylor expanded in x around inf 95.2%

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

      1. associate-/l* 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in z around 0 71.5%

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

   if -10.5 < x < 6.4999999999999996e-90 or 2.9499999999999998e-37 < x < 4

   1. Initial program 96.3%

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

   2. Taylor expanded in x around 0 80.6%

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

   if 6.4999999999999996e-90 < x < 2.9499999999999998e-37 or 1.45000000000000003e162 < x

   1. Initial program 83.6%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in x around inf 84.2%

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

   5. Taylor expanded in z around inf 61.4%

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

      1. associate-*r/ 81.8%

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

   7. Simplified 81.8%

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

   if 1.95e49 < x < 2.99999999999999997e81

   1. Initial program 99.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 99.4%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-37}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+81}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+162}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-33}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+81}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (/ 4.0 y))))
   (if (<= x -10.2)
     t_0
     (if (<= x 6.5e-90)
       t_1
       (if (<= x 2.85e-33)
         (fabs (* z (/ x y)))
         (if (<= x 4.0)
           t_1
           (if (<= x 1.7e+49)
             t_0
             (if (<= x 2.05e+81)
               (fabs (* x (/ z y)))
               (if (<= x 7.2e+155) t_0 (fabs (/ z (/ y x))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((4.0 / y));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 6.5e-90) {
		tmp = t_1;
	} else if (x <= 2.85e-33) {
		tmp = fabs((z * (x / y)));
	} else if (x <= 4.0) {
		tmp = t_1;
	} else if (x <= 1.7e+49) {
		tmp = t_0;
	} else if (x <= 2.05e+81) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 7.2e+155) {
		tmp = t_0;
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

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((4.0d0 / y))
    if (x <= (-10.2d0)) then
        tmp = t_0
    else if (x <= 6.5d-90) then
        tmp = t_1
    else if (x <= 2.85d-33) then
        tmp = abs((z * (x / y)))
    else if (x <= 4.0d0) then
        tmp = t_1
    else if (x <= 1.7d+49) then
        tmp = t_0
    else if (x <= 2.05d+81) then
        tmp = abs((x * (z / y)))
    else if (x <= 7.2d+155) then
        tmp = t_0
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((4.0 / y));
	double tmp;
	if (x <= -10.2) {
		tmp = t_0;
	} else if (x <= 6.5e-90) {
		tmp = t_1;
	} else if (x <= 2.85e-33) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= 4.0) {
		tmp = t_1;
	} else if (x <= 1.7e+49) {
		tmp = t_0;
	} else if (x <= 2.05e+81) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 7.2e+155) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -10.2:
		tmp = t_0
	elif x <= 6.5e-90:
		tmp = t_1
	elif x <= 2.85e-33:
		tmp = math.fabs((z * (x / y)))
	elif x <= 4.0:
		tmp = t_1
	elif x <= 1.7e+49:
		tmp = t_0
	elif x <= 2.05e+81:
		tmp = math.fabs((x * (z / y)))
	elif x <= 7.2e+155:
		tmp = t_0
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 6.5e-90)
		tmp = t_1;
	elseif (x <= 2.85e-33)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= 4.0)
		tmp = t_1;
	elseif (x <= 1.7e+49)
		tmp = t_0;
	elseif (x <= 2.05e+81)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 7.2e+155)
		tmp = t_0;
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -10.2)
		tmp = t_0;
	elseif (x <= 6.5e-90)
		tmp = t_1;
	elseif (x <= 2.85e-33)
		tmp = abs((z * (x / y)));
	elseif (x <= 4.0)
		tmp = t_1;
	elseif (x <= 1.7e+49)
		tmp = t_0;
	elseif (x <= 2.05e+81)
		tmp = abs((x * (z / y)));
	elseif (x <= 7.2e+155)
		tmp = t_0;
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.2], t$95$0, If[LessEqual[x, 6.5e-90], t$95$1, If[LessEqual[x, 2.85e-33], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], t$95$1, If[LessEqual[x, 1.7e+49], t$95$0, If[LessEqual[x, 2.05e+81], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 7.2e+155], t$95$0, N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -10.199999999999999 or 4 < x < 1.7e49 or 2.05000000000000006e81 < x < 7.20000000000000015e155

   1. Initial program 83.5%

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

      1. associate-*l/ 86.3%

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

      2. sub-div 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in x around inf 96.2%

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

      1. associate-/l* 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in z around 0 71.7%

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

   if -10.199999999999999 < x < 6.4999999999999996e-90 or 2.85000000000000013e-33 < x < 4

   1. Initial program 96.3%

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

   2. Taylor expanded in x around 0 80.6%

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

   if 6.4999999999999996e-90 < x < 2.85000000000000013e-33

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 97.3%

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

   5. Taylor expanded in z around inf 97.3%

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

      1. associate-*r/ 97.0%

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

   7. Simplified 97.0%

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

   if 1.7e49 < x < 2.05000000000000006e81

   1. Initial program 99.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 99.4%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 7.20000000000000015e155 < x

   1. Initial program 82.7%

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

   3. Simplified 80.8%

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

      1. associate-*l/ 81.0%

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

      2. associate-/l* 80.8%

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

      3. fma-udef 80.8%

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

      4. associate-+r- 80.8%

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

      5. fma-udef 80.8%

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

   5. Applied egg-rr 80.8%

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

   6. Taylor expanded in z around inf 51.4%

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

      1. associate-/l* 78.5%

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

   8. Simplified 78.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-33}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+81}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 4: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+81}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (/ 4.0 y))))
   (if (<= x -10.5)
     t_0
     (if (<= x 6.5e-90)
       t_1
       (if (<= x 1.6e-26)
         (fabs (/ (* x z) y))
         (if (<= x 4.0)
           t_1
           (if (<= x 1.75e+48)
             t_0
             (if (<= x 2.05e+81)
               (fabs (* x (/ z y)))
               (if (<= x 1.75e+155) t_0 (fabs (/ z (/ y x))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((4.0 / y));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 6.5e-90) {
		tmp = t_1;
	} else if (x <= 1.6e-26) {
		tmp = fabs(((x * z) / y));
	} else if (x <= 4.0) {
		tmp = t_1;
	} else if (x <= 1.75e+48) {
		tmp = t_0;
	} else if (x <= 2.05e+81) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 1.75e+155) {
		tmp = t_0;
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

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((4.0d0 / y))
    if (x <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 6.5d-90) then
        tmp = t_1
    else if (x <= 1.6d-26) then
        tmp = abs(((x * z) / y))
    else if (x <= 4.0d0) then
        tmp = t_1
    else if (x <= 1.75d+48) then
        tmp = t_0
    else if (x <= 2.05d+81) then
        tmp = abs((x * (z / y)))
    else if (x <= 1.75d+155) then
        tmp = t_0
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((4.0 / y));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 6.5e-90) {
		tmp = t_1;
	} else if (x <= 1.6e-26) {
		tmp = Math.abs(((x * z) / y));
	} else if (x <= 4.0) {
		tmp = t_1;
	} else if (x <= 1.75e+48) {
		tmp = t_0;
	} else if (x <= 2.05e+81) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 1.75e+155) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -10.5:
		tmp = t_0
	elif x <= 6.5e-90:
		tmp = t_1
	elif x <= 1.6e-26:
		tmp = math.fabs(((x * z) / y))
	elif x <= 4.0:
		tmp = t_1
	elif x <= 1.75e+48:
		tmp = t_0
	elif x <= 2.05e+81:
		tmp = math.fabs((x * (z / y)))
	elif x <= 1.75e+155:
		tmp = t_0
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 6.5e-90)
		tmp = t_1;
	elseif (x <= 1.6e-26)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (x <= 4.0)
		tmp = t_1;
	elseif (x <= 1.75e+48)
		tmp = t_0;
	elseif (x <= 2.05e+81)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 1.75e+155)
		tmp = t_0;
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 6.5e-90)
		tmp = t_1;
	elseif (x <= 1.6e-26)
		tmp = abs(((x * z) / y));
	elseif (x <= 4.0)
		tmp = t_1;
	elseif (x <= 1.75e+48)
		tmp = t_0;
	elseif (x <= 2.05e+81)
		tmp = abs((x * (z / y)));
	elseif (x <= 1.75e+155)
		tmp = t_0;
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 6.5e-90], t$95$1, If[LessEqual[x, 1.6e-26], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], t$95$1, If[LessEqual[x, 1.75e+48], t$95$0, If[LessEqual[x, 2.05e+81], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.75e+155], t$95$0, N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -10.5 or 4 < x < 1.7499999999999999e48 or 2.05000000000000006e81 < x < 1.74999999999999992e155

   1. Initial program 83.5%

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

      1. associate-*l/ 86.3%

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

      2. sub-div 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in x around inf 96.2%

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

      1. associate-/l* 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in z around 0 71.7%

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

   if -10.5 < x < 6.4999999999999996e-90 or 1.6000000000000001e-26 < x < 4

   1. Initial program 96.3%

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

   2. Taylor expanded in x around 0 80.6%

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

   if 6.4999999999999996e-90 < x < 1.6000000000000001e-26

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around inf 97.3%

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

   if 1.7499999999999999e48 < x < 2.05000000000000006e81

   1. Initial program 99.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 99.4%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 1.74999999999999992e155 < x

   1. Initial program 82.7%

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

   3. Simplified 80.8%

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

      1. associate-*l/ 81.0%

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

      2. associate-/l* 80.8%

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

      3. fma-udef 80.8%

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

      4. associate-+r- 80.8%

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

      5. fma-udef 80.8%

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

   5. Applied egg-rr 80.8%

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

   6. Taylor expanded in z around inf 51.4%

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

      1. associate-/l* 78.5%

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

   8. Simplified 78.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+48}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+81}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+24} \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\left(z + -1\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot z - 4}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e+24) (not (<= x 4.2)))
   (fabs (* (+ z -1.0) (/ x y)))
   (fabs (/ 1.0 (/ y (- (* x z) 4.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e+24) || !(x <= 4.2)) {
		tmp = fabs(((z + -1.0) * (x / y)));
	} else {
		tmp = fabs((1.0 / (y / ((x * z) - 4.0))));
	}
	return tmp;
}

Fortran

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 <= (-1.15d+24)) .or. (.not. (x <= 4.2d0))) then
        tmp = abs(((z + (-1.0d0)) * (x / y)))
    else
        tmp = abs((1.0d0 / (y / ((x * z) - 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e+24) || !(x <= 4.2)) {
		tmp = Math.abs(((z + -1.0) * (x / y)));
	} else {
		tmp = Math.abs((1.0 / (y / ((x * z) - 4.0))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e+24) or not (x <= 4.2):
		tmp = math.fabs(((z + -1.0) * (x / y)))
	else:
		tmp = math.fabs((1.0 / (y / ((x * z) - 4.0))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e+24) || !(x <= 4.2))
		tmp = abs(Float64(Float64(z + -1.0) * Float64(x / y)));
	else
		tmp = abs(Float64(1.0 / Float64(y / Float64(Float64(x * z) - 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.15e+24) || ~((x <= 4.2)))
		tmp = abs(((z + -1.0) * (x / y)));
	else
		tmp = abs((1.0 / (y / ((x * z) - 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e+24], N[Not[LessEqual[x, 4.2]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(1.0 / N[(y / N[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e24 or 4.20000000000000018 < x

   1. Initial program 83.5%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in x around inf 91.7%

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

      1. associate-/l* 98.8%

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

      2. div-inv 98.7%

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

      3. sub-neg 98.7%

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

      4. metadata-eval 98.7%

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

      5. clear-num 99.0%

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

   6. Applied egg-rr 99.0%

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

   if -1.15e24 < x < 4.20000000000000018

   1. Initial program 96.5%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. associate-/l* 99.8%

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

      3. fma-udef 99.8%

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

      4. associate-+r- 99.8%

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

      5. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

   7. Taylor expanded in z around inf 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+24} \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\left(z + -1\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot z - 4}}\right|\\ \end{array} \]

Alternative 6: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 26000000:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+255}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -10.5)
     t_0
     (if (<= x 26000000.0)
       (fabs (/ 4.0 y))
       (if (<= x 2.9e+255) (fabs (* x (/ z y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 26000000.0) {
		tmp = fabs((4.0 / y));
	} else if (x <= 2.9e+255) {
		tmp = fabs((x * (z / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 26000000.0d0) then
        tmp = abs((4.0d0 / y))
    else if (x <= 2.9d+255) then
        tmp = abs((x * (z / y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 26000000.0) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 2.9e+255) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -10.5:
		tmp = t_0
	elif x <= 26000000.0:
		tmp = math.fabs((4.0 / y))
	elif x <= 2.9e+255:
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 26000000.0)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 2.9e+255)
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 26000000.0)
		tmp = abs((4.0 / y));
	elseif (x <= 2.9e+255)
		tmp = abs((x * (z / y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 26000000.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.9e+255], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.5 or 2.9000000000000002e255 < x

   1. Initial program 78.8%

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

      1. associate-*l/ 80.6%

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

      2. sub-div 95.8%

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

   3. Applied egg-rr 95.8%

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

   4. Taylor expanded in x around inf 95.7%

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

      1. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 77.1%

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

   if -10.5 < x < 2.6e7

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 77.0%

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

   if 2.6e7 < x < 2.9000000000000002e255

   1. Initial program 90.0%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in z around inf 52.0%

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

      1. associate-*l/ 63.0%

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

      2. *-commutative 63.0%

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

   6. Simplified 63.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 26000000:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+255}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2e+127)
   (fabs (/ (* x z) y))
   (if (<= z 2e+34) (fabs (+ (/ x y) (/ 4.0 y))) (fabs (/ z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+127) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 2e+34) {
		tmp = fabs(((x / y) + (4.0 / y)));
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

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 <= (-3.2d+127)) then
        tmp = abs(((x * z) / y))
    else if (z <= 2d+34) then
        tmp = abs(((x / y) + (4.0d0 / y)))
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+127) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 2e+34) {
		tmp = Math.abs(((x / y) + (4.0 / y)));
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.2e+127:
		tmp = math.fabs(((x * z) / y))
	elif z <= 2e+34:
		tmp = math.fabs(((x / y) + (4.0 / y)))
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2e+127)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 2e+34)
		tmp = abs(Float64(Float64(x / y) + Float64(4.0 / y)));
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.2e+127)
		tmp = abs(((x * z) / y));
	elseif (z <= 2e+34)
		tmp = abs(((x / y) + (4.0 / y)));
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2e+127], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2e+34], N[Abs[N[(N[(x / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999976e127

   1. Initial program 93.3%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in z around inf 81.8%

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

   if -3.19999999999999976e127 < z < 1.99999999999999989e34

   1. Initial program 94.3%

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

   2. Taylor expanded in z around 0 95.0%

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

      1. associate-*r/ 95.0%

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

      2. metadata-eval 95.0%

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

   4. Simplified 95.0%

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

   if 1.99999999999999989e34 < z

   1. Initial program 77.5%

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

   3. Simplified 92.8%

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

      1. associate-*l/ 92.8%

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

      2. associate-/l* 92.7%

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

      3. fma-udef 92.7%

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

      4. associate-+r- 92.7%

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

      5. fma-udef 92.7%

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

   5. Applied egg-rr 92.7%

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

   6. Taylor expanded in z around inf 74.1%

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

      1. associate-/l* 79.1%

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

   8. Simplified 79.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 8: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.08e+129)
   (fabs (/ (* x z) y))
   (if (<= z 3.9e+20)
     (fabs (+ (/ x y) (/ 4.0 y)))
     (fabs (/ (+ z -1.0) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.08e+129) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 3.9e+20) {
		tmp = fabs(((x / y) + (4.0 / y)));
	} else {
		tmp = fabs(((z + -1.0) / (y / x)));
	}
	return tmp;
}

Fortran

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 <= (-1.08d+129)) then
        tmp = abs(((x * z) / y))
    else if (z <= 3.9d+20) then
        tmp = abs(((x / y) + (4.0d0 / y)))
    else
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.08e+129) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 3.9e+20) {
		tmp = Math.abs(((x / y) + (4.0 / y)));
	} else {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.08e+129:
		tmp = math.fabs(((x * z) / y))
	elif z <= 3.9e+20:
		tmp = math.fabs(((x / y) + (4.0 / y)))
	else:
		tmp = math.fabs(((z + -1.0) / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.08e+129)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 3.9e+20)
		tmp = abs(Float64(Float64(x / y) + Float64(4.0 / y)));
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.08e+129)
		tmp = abs(((x * z) / y));
	elseif (z <= 3.9e+20)
		tmp = abs(((x / y) + (4.0 / y)));
	else
		tmp = abs(((z + -1.0) / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.08e+129], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.9e+20], N[Abs[N[(N[(x / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.08e129

   1. Initial program 93.3%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in z around inf 81.8%

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

   if -1.08e129 < z < 3.9e20

   1. Initial program 94.3%

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

   2. Taylor expanded in z around 0 95.0%

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

      1. associate-*r/ 95.0%

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

      2. metadata-eval 95.0%

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

   4. Simplified 95.0%

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

   if 3.9e20 < z

   1. Initial program 77.5%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around inf 74.1%

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

      1. associate-/l* 79.1%

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

      2. sub-neg 79.1%

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

      3. metadata-eval 79.1%

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

   6. Simplified 79.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 9: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(z + -1\right) \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.4e+16)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (* (+ z -1.0) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.4e+16) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((z + -1.0) * (x / y)));
	}
	return tmp;
}

Fortran

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 <= 3.4d+16) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((z + (-1.0d0)) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.4e+16) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((z + -1.0) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.4e+16:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((z + -1.0) * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.4e+16)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(z + -1.0) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.4e+16)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((z + -1.0) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.4e+16], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4e16

   1. Initial program 92.1%

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

      1. associate-*l/ 95.2%

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

      2. sub-div 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 3.4e16 < x

   1. Initial program 86.2%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around inf 89.8%

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

      1. associate-/l* 99.8%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(z + -1\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 10: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2e+127)
   (fabs (/ (* x z) y))
   (if (<= z 2.1e+24) (fabs (/ (- -4.0 x) y)) (fabs (/ z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+127) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 2.1e+24) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

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 <= (-3.2d+127)) then
        tmp = abs(((x * z) / y))
    else if (z <= 2.1d+24) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+127) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 2.1e+24) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.2e+127:
		tmp = math.fabs(((x * z) / y))
	elif z <= 2.1e+24:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2e+127)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 2.1e+24)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.2e+127)
		tmp = abs(((x * z) / y));
	elseif (z <= 2.1e+24)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2e+127], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.1e+24], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999976e127

   1. Initial program 93.3%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in z around inf 81.8%

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

   if -3.19999999999999976e127 < z < 2.1000000000000001e24

   1. Initial program 94.3%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in z around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. distribute-lft-in 94.9%

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

      3. metadata-eval 94.9%

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

      4. neg-mul-1 94.9%

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

      5. sub-neg 94.9%

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

   6. Simplified 94.9%

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

   if 2.1000000000000001e24 < z

   1. Initial program 77.5%

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

   3. Simplified 92.8%

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

      1. associate-*l/ 92.8%

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

      2. associate-/l* 92.7%

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

      3. fma-udef 92.7%

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

      4. associate-+r- 92.7%

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

      5. fma-udef 92.7%

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

   5. Applied egg-rr 92.7%

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

   6. Taylor expanded in z around inf 74.1%

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

      1. associate-/l* 79.1%

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

   8. Simplified 79.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 11: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

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 <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -10.5], 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 < -10.5 or 4 < x

   1. Initial program 84.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. sub-div 92.8%

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

   3. Applied egg-rr 92.8%

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

   4. Taylor expanded in x around inf 91.9%

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

      1. associate-/l* 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in z around 0 65.5%

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

   if -10.5 < x < 4

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 77.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 12: 39.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in x around 0 44.1%

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

3. Final simplification 44.1%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))