fabs fraction 1

Percentage Accurate: 91.2% → 99.8%

Time: 6.1s

Alternatives: 7

Speedup: 1.0×

• 20.8% 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: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+21)
   (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 <= 5e+21) {
		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 <= 5e+21)
		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, 5e+21], 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 < 5e21

   1. Initial program 90.6%

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

      1. associate-*l/ 90.7%

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

      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%

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

   if 5e21 < y

   1. Initial program 94.5%

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

      1. fabs-sub 94.5%

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

      2. associate-*l/ 91.0%

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

      3. *-commutative 91.0%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ t_2 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;z \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* z (/ x y)))) (t_2 (fabs (/ 4.0 y))))
   (if (<= z -1e+95)
     t_1
     (if (<= z -4.5e-28)
       t_2
       (if (<= z 3.9e-259)
         t_0
         (if (<= z 7.5e-116)
           t_2
           (if (<= z 2.3e-12) t_0 (if (<= z 3.2e+60) t_2 t_1))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((z * (x / y)));
	double t_2 = fabs((4.0 / y));
	double tmp;
	if (z <= -1e+95) {
		tmp = t_1;
	} else if (z <= -4.5e-28) {
		tmp = t_2;
	} else if (z <= 3.9e-259) {
		tmp = t_0;
	} else if (z <= 7.5e-116) {
		tmp = t_2;
	} else if (z <= 2.3e-12) {
		tmp = t_0;
	} else if (z <= 3.2e+60) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((z * (x / y)))
    t_2 = abs((4.0d0 / y))
    if (z <= (-1d+95)) then
        tmp = t_1
    else if (z <= (-4.5d-28)) then
        tmp = t_2
    else if (z <= 3.9d-259) then
        tmp = t_0
    else if (z <= 7.5d-116) then
        tmp = t_2
    else if (z <= 2.3d-12) then
        tmp = t_0
    else if (z <= 3.2d+60) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((z * (x / y)));
	double t_2 = Math.abs((4.0 / y));
	double tmp;
	if (z <= -1e+95) {
		tmp = t_1;
	} else if (z <= -4.5e-28) {
		tmp = t_2;
	} else if (z <= 3.9e-259) {
		tmp = t_0;
	} else if (z <= 7.5e-116) {
		tmp = t_2;
	} else if (z <= 2.3e-12) {
		tmp = t_0;
	} else if (z <= 3.2e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((z * (x / y)))
	t_2 = math.fabs((4.0 / y))
	tmp = 0
	if z <= -1e+95:
		tmp = t_1
	elif z <= -4.5e-28:
		tmp = t_2
	elif z <= 3.9e-259:
		tmp = t_0
	elif z <= 7.5e-116:
		tmp = t_2
	elif z <= 2.3e-12:
		tmp = t_0
	elif z <= 3.2e+60:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(z * Float64(x / y)))
	t_2 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (z <= -1e+95)
		tmp = t_1;
	elseif (z <= -4.5e-28)
		tmp = t_2;
	elseif (z <= 3.9e-259)
		tmp = t_0;
	elseif (z <= 7.5e-116)
		tmp = t_2;
	elseif (z <= 2.3e-12)
		tmp = t_0;
	elseif (z <= 3.2e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((z * (x / y)));
	t_2 = abs((4.0 / y));
	tmp = 0.0;
	if (z <= -1e+95)
		tmp = t_1;
	elseif (z <= -4.5e-28)
		tmp = t_2;
	elseif (z <= 3.9e-259)
		tmp = t_0;
	elseif (z <= 7.5e-116)
		tmp = t_2;
	elseif (z <= 2.3e-12)
		tmp = t_0;
	elseif (z <= 3.2e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1e+95], t$95$1, If[LessEqual[z, -4.5e-28], t$95$2, If[LessEqual[z, 3.9e-259], t$95$0, If[LessEqual[z, 7.5e-116], t$95$2, If[LessEqual[z, 2.3e-12], t$95$0, If[LessEqual[z, 3.2e+60], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000002e95 or 3.19999999999999991e60 < z

   1. Initial program 88.5%

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

   2. Taylor expanded in z around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-/l* 84.4%

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

      3. associate-/r/ 82.8%

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

      4. associate-*r* 82.8%

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

      5. *-commutative 82.8%

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

      6. mul-1-neg 82.8%

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

      7. distribute-frac-neg 82.8%

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

   4. Simplified 82.8%

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

      1. expm1-log1p-u 39.0%

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

      2. expm1-udef 31.1%

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

      3. add-sqr-sqrt 15.0%

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

      4. sqrt-unprod 28.2%

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

      5. sqr-neg 28.2%

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

      6. sqrt-unprod 18.7%

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

      7. add-sqr-sqrt 42.0%

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

   6. Applied egg-rr 42.0%

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

      1. expm1-def 49.9%

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

      2. expm1-log1p 82.8%

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

   8. Simplified 82.8%

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

   if -1.00000000000000002e95 < z < -4.4999999999999998e-28 or 3.90000000000000016e-259 < z < 7.5000000000000004e-116 or 2.29999999999999989e-12 < z < 3.19999999999999991e60

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 61.6%

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

   if -4.4999999999999998e-28 < z < 3.90000000000000016e-259 or 7.5000000000000004e-116 < z < 2.29999999999999989e-12

   1. Initial program 95.0%

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

   2. Taylor expanded in z around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 69.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-259}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;z \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (/ 4.0 y))))
   (if (<= z -1e+95)
     (fabs (* z (/ x y)))
     (if (<= z -4.5e-27)
       t_1
       (if (<= z 3.4e-260)
         t_0
         (if (<= z 5.2e-117)
           t_1
           (if (<= z 2.7e-12)
             t_0
             (if (<= z 3.2e+60) t_1 (fabs (* x (/ z y)))))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((4.0 / y));
	double tmp;
	if (z <= -1e+95) {
		tmp = fabs((z * (x / y)));
	} else if (z <= -4.5e-27) {
		tmp = t_1;
	} else if (z <= 3.4e-260) {
		tmp = t_0;
	} else if (z <= 5.2e-117) {
		tmp = t_1;
	} else if (z <= 2.7e-12) {
		tmp = t_0;
	} else if (z <= 3.2e+60) {
		tmp = t_1;
	} else {
		tmp = fabs((x * (z / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((4.0d0 / y))
    if (z <= (-1d+95)) then
        tmp = abs((z * (x / y)))
    else if (z <= (-4.5d-27)) then
        tmp = t_1
    else if (z <= 3.4d-260) then
        tmp = t_0
    else if (z <= 5.2d-117) then
        tmp = t_1
    else if (z <= 2.7d-12) then
        tmp = t_0
    else if (z <= 3.2d+60) then
        tmp = t_1
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((4.0 / y));
	double tmp;
	if (z <= -1e+95) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= -4.5e-27) {
		tmp = t_1;
	} else if (z <= 3.4e-260) {
		tmp = t_0;
	} else if (z <= 5.2e-117) {
		tmp = t_1;
	} else if (z <= 2.7e-12) {
		tmp = t_0;
	} else if (z <= 3.2e+60) {
		tmp = t_1;
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((4.0 / y))
	tmp = 0
	if z <= -1e+95:
		tmp = math.fabs((z * (x / y)))
	elif z <= -4.5e-27:
		tmp = t_1
	elif z <= 3.4e-260:
		tmp = t_0
	elif z <= 5.2e-117:
		tmp = t_1
	elif z <= 2.7e-12:
		tmp = t_0
	elif z <= 3.2e+60:
		tmp = t_1
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (z <= -1e+95)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= -4.5e-27)
		tmp = t_1;
	elseif (z <= 3.4e-260)
		tmp = t_0;
	elseif (z <= 5.2e-117)
		tmp = t_1;
	elseif (z <= 2.7e-12)
		tmp = t_0;
	elseif (z <= 3.2e+60)
		tmp = t_1;
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((4.0 / y));
	tmp = 0.0;
	if (z <= -1e+95)
		tmp = abs((z * (x / y)));
	elseif (z <= -4.5e-27)
		tmp = t_1;
	elseif (z <= 3.4e-260)
		tmp = t_0;
	elseif (z <= 5.2e-117)
		tmp = t_1;
	elseif (z <= 2.7e-12)
		tmp = t_0;
	elseif (z <= 3.2e+60)
		tmp = t_1;
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := 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[z, -1e+95], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -4.5e-27], t$95$1, If[LessEqual[z, 3.4e-260], t$95$0, If[LessEqual[z, 5.2e-117], t$95$1, If[LessEqual[z, 2.7e-12], t$95$0, If[LessEqual[z, 3.2e+60], t$95$1, N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.00000000000000002e95

   1. Initial program 96.3%

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

   2. Taylor expanded in z around inf 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/l* 81.2%

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

      3. associate-/r/ 81.6%

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

      4. associate-*r* 81.6%

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

      5. *-commutative 81.6%

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

      6. mul-1-neg 81.6%

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

      7. distribute-frac-neg 81.6%

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

   4. Simplified 81.6%

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

      1. expm1-log1p-u 35.9%

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

      2. expm1-udef 28.0%

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

      3. add-sqr-sqrt 13.6%

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

      4. sqrt-unprod 25.4%

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

      5. sqr-neg 25.4%

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

      6. sqrt-unprod 18.4%

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

      7. add-sqr-sqrt 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 51.8%

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

      2. expm1-log1p 81.6%

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

   8. Simplified 81.6%

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

   if -1.00000000000000002e95 < z < -4.5000000000000002e-27 or 3.3999999999999998e-260 < z < 5.19999999999999966e-117 or 2.6999999999999998e-12 < z < 3.19999999999999991e60

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 61.6%

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

   if -4.5000000000000002e-27 < z < 3.3999999999999998e-260 or 5.19999999999999966e-117 < z < 2.6999999999999998e-12

   1. Initial program 95.0%

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

   2. Taylor expanded in z around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 69.1%

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

   if 3.19999999999999991e60 < z

   1. Initial program 79.6%

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

      1. fabs-sub 79.6%

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

      2. associate-*l/ 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-*l/ 77.7%

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

      5. *-commutative 77.7%

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

      6. fma-neg 81.9%

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

      7. distribute-neg-frac 81.9%

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

      8. +-commutative 81.9%

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

      9. distribute-neg-in 81.9%

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

      10. unsub-neg 81.9%

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

      11. metadata-eval 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around inf 88.2%

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

   5. Taylor expanded in z around inf 88.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-260}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+48}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+48)
   (fabs (* (/ x y) (- 1.0 z)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+48) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3d+48)) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+48) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= -3e+48:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= -3e+48)
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e+48)
		tmp = abs(((x / y) * (1.0 - z)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, -3e+48], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3e48

   1. Initial program 83.9%

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

      1. fabs-neg 83.9%

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

      2. sub-neg 83.9%

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

      3. distribute-neg-in 83.9%

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

      4. sub-neg 83.9%

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

      5. distribute-neg-frac 83.9%

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

      6. associate-*l/ 72.8%

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

      7. distribute-neg-frac 72.8%

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

      8. neg-mul-1 72.8%

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

      9. associate-*l/ 72.7%

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

      10. neg-mul-1 72.7%

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

      11. associate-*l/ 72.7%

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

      12. distribute-lft-out-- 86.7%

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

      13. fabs-mul 86.7%

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

      14. fabs-sub 86.7%

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

      15. fabs-mul 86.7%

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

      16. associate-*l/ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around inf 86.8%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -3e48 < x

   1. Initial program 93.2%

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

      1. associate-*l/ 95.1%

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

      2. sub-div 98.6%

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

   3. Applied egg-rr 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+48}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.4e+106)
   (fabs (* z (/ x y)))
   (if (<= z 1.3e+61) (fabs (/ (- -4.0 x) y)) (fabs (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+106) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 1.3e+61) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((x * (z / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.4d+106)) then
        tmp = abs((z * (x / y)))
    else if (z <= 1.3d+61) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+106) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 1.3e+61) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -1.4e+106:
		tmp = math.fabs((z * (x / y)))
	elif z <= 1.3e+61:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.4e+106)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 1.3e+61)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.4e+106)
		tmp = abs((z * (x / y)));
	elseif (z <= 1.3e+61)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -1.4e+106], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.3e+61], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.39999999999999996e106

   1. Initial program 96.2%

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

   2. Taylor expanded in z around inf 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-/l* 82.4%

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

      3. associate-/r/ 82.7%

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

      4. associate-*r* 82.7%

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

      5. *-commutative 82.7%

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

      6. mul-1-neg 82.7%

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

      7. distribute-frac-neg 82.7%

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

   4. Simplified 82.7%

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

      1. expm1-log1p-u 35.6%

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

      2. expm1-udef 27.4%

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

      3. add-sqr-sqrt 14.2%

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

      4. sqrt-unprod 26.3%

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

      5. sqr-neg 26.3%

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

      6. sqrt-unprod 19.1%

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

      7. add-sqr-sqrt 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. expm1-def 53.7%

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

      2. expm1-log1p 82.7%

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

   8. Simplified 82.7%

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

   if -1.39999999999999996e106 < z < 1.29999999999999986e61

   1. Initial program 93.5%

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

      1. fabs-sub 93.5%

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

      2. associate-*l/ 93.6%

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

      3. *-commutative 93.6%

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

      4. associate-*l/ 93.0%

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

      5. *-commutative 93.0%

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

      6. fma-neg 98.1%

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

      7. distribute-neg-frac 98.1%

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

      8. +-commutative 98.1%

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

      9. distribute-neg-in 98.1%

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

      10. unsub-neg 98.1%

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

      11. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in z around 0 92.2%

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

      1. associate-*r/ 92.2%

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

      2. distribute-lft-in 92.2%

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

      3. metadata-eval 92.2%

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

      4. neg-mul-1 92.2%

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

      5. sub-neg 92.2%

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

   6. Simplified 92.2%

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

   if 1.29999999999999986e61 < z

   1. Initial program 79.6%

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

      1. fabs-sub 79.6%

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

      2. associate-*l/ 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-*l/ 77.7%

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

      5. *-commutative 77.7%

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

      6. fma-neg 81.9%

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

      7. distribute-neg-frac 81.9%

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

      8. +-commutative 81.9%

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

      9. distribute-neg-in 81.9%

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

      10. unsub-neg 81.9%

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

      11. metadata-eval 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in x around inf 88.2%

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

   5. Taylor expanded in z around inf 88.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+61}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 6: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.2) (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 <= -10.2) || !(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 <= (-10.2d0)) .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 <= -10.2) || !(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 <= -10.2) 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 <= -10.2) || !(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 <= -10.2) || ~((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, -10.2], 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 < -10.199999999999999 or 4 < x

   1. Initial program 87.9%

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

   2. Taylor expanded in z around 0 63.2%

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

      1. associate-*r/ 63.2%

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

      2. metadata-eval 63.2%

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

   4. Simplified 63.2%

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

   5. Taylor expanded in x around inf 62.1%

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

   if -10.199999999999999 < x < 4

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 66.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 7: 39.3% 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 91.4%

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

2. Taylor expanded in x around 0 34.3%

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

3. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2023223 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))