fabs fraction 1

Percentage Accurate: 92.0% → 99.4%

Time: 6.2s

Alternatives: 11

Speedup: 1.0×

• 20.3% 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: 92.0% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 4 \cdot 10^{-91}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-91)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-91) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-91)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4e-91], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000009e-91

   1. Initial program 90.6%

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

   2. Taylor expanded in y around 0 96.6%

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

   if 4.00000000000000009e-91 < y

   1. Initial program 94.3%

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

   3. Simplified 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-91}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := z \cdot \frac{x}{y_m}\\ t_1 := \left|t_0 - \frac{4 + x}{y_m}\right|\\ \mathbf{if}\;t_1 \leq 10^{-181}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (* z (/ x y_m))) (t_1 (fabs (- t_0 (/ (+ 4.0 x) y_m)))))
   (if (<= t_1 1e-181)
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
     (if (<= t_1 1e+304) t_1 (fabs t_0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = z * (x / y_m);
	double t_1 = fabs((t_0 - ((4.0 + x) / y_m)));
	double tmp;
	if (t_1 <= 1e-181) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * (x / y_m)
    t_1 = abs((t_0 - ((4.0d0 + x) / y_m)))
    if (t_1 <= 1d-181) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else if (t_1 <= 1d+304) then
        tmp = t_1
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = z * (x / y_m);
	double t_1 = Math.abs((t_0 - ((4.0 + x) / y_m)));
	double tmp;
	if (t_1 <= 1e-181) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = z * (x / y_m)
	t_1 = math.fabs((t_0 - ((4.0 + x) / y_m)))
	tmp = 0
	if t_1 <= 1e-181:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	elif t_1 <= 1e+304:
		tmp = t_1
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(z * Float64(x / y_m))
	t_1 = abs(Float64(t_0 - Float64(Float64(4.0 + x) / y_m)))
	tmp = 0.0
	if (t_1 <= 1e-181)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = z * (x / y_m);
	t_1 = abs((t_0 - ((4.0 + x) / y_m)));
	tmp = 0.0;
	if (t_1 <= 1e-181)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(t$95$0 - N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1e-181], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+304], t$95$1, N[Abs[t$95$0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 1.00000000000000005e-181

   1. Initial program 79.9%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 1.00000000000000005e-181 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 9.9999999999999994e303

   1. Initial program 99.9%

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

   if 9.9999999999999994e303 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 72.2%

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

   2. Taylor expanded in z around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. associate-*l/ 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

   4. Simplified 100.0%

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

      1. expm1-log1p-u 59.0%

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

      2. expm1-udef 59.0%

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

      3. add-sqr-sqrt 25.9%

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

      4. sqrt-unprod 40.7%

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

      5. sqr-neg 40.7%

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

      6. sqrt-unprod 24.1%

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

      7. add-sqr-sqrt 40.7%

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

      8. associate-*l/ 37.2%

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

      9. *-commutative 37.2%

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

      10. associate-/l* 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. expm1-def 40.7%

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

      2. expm1-log1p 100.0%

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

   8. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|z \cdot \frac{x}{y} - \frac{4 + x}{y}\right| \leq 10^{-181}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{elif}\;\left|z \cdot \frac{x}{y} - \frac{4 + x}{y}\right| \leq 10^{+304}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{4}{y_m}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+99} \lor \neg \left(x \leq 2.8 \cdot 10^{+186}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ 4.0 y_m))))
   (if (<= x -2.5e-16)
     t_0
     (if (<= x 1.55e-121)
       t_1
       (if (<= x 8.5e-61)
         t_0
         (if (<= x 5.6e-20)
           t_1
           (if (or (<= x 1.75e+99) (not (<= x 2.8e+186)))
             t_0
             (fabs (/ x y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((4.0 / y_m));
	double tmp;
	if (x <= -2.5e-16) {
		tmp = t_0;
	} else if (x <= 1.55e-121) {
		tmp = t_1;
	} else if (x <= 8.5e-61) {
		tmp = t_0;
	} else if (x <= 5.6e-20) {
		tmp = t_1;
	} else if ((x <= 1.75e+99) || !(x <= 2.8e+186)) {
		tmp = t_0;
	} else {
		tmp = fabs((x / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((4.0d0 / y_m))
    if (x <= (-2.5d-16)) then
        tmp = t_0
    else if (x <= 1.55d-121) then
        tmp = t_1
    else if (x <= 8.5d-61) then
        tmp = t_0
    else if (x <= 5.6d-20) then
        tmp = t_1
    else if ((x <= 1.75d+99) .or. (.not. (x <= 2.8d+186))) then
        tmp = t_0
    else
        tmp = abs((x / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((4.0 / y_m));
	double tmp;
	if (x <= -2.5e-16) {
		tmp = t_0;
	} else if (x <= 1.55e-121) {
		tmp = t_1;
	} else if (x <= 8.5e-61) {
		tmp = t_0;
	} else if (x <= 5.6e-20) {
		tmp = t_1;
	} else if ((x <= 1.75e+99) || !(x <= 2.8e+186)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((4.0 / y_m))
	tmp = 0
	if x <= -2.5e-16:
		tmp = t_0
	elif x <= 1.55e-121:
		tmp = t_1
	elif x <= 8.5e-61:
		tmp = t_0
	elif x <= 5.6e-20:
		tmp = t_1
	elif (x <= 1.75e+99) or not (x <= 2.8e+186):
		tmp = t_0
	else:
		tmp = math.fabs((x / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(4.0 / y_m))
	tmp = 0.0
	if (x <= -2.5e-16)
		tmp = t_0;
	elseif (x <= 1.55e-121)
		tmp = t_1;
	elseif (x <= 8.5e-61)
		tmp = t_0;
	elseif (x <= 5.6e-20)
		tmp = t_1;
	elseif ((x <= 1.75e+99) || !(x <= 2.8e+186))
		tmp = t_0;
	else
		tmp = abs(Float64(x / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((4.0 / y_m));
	tmp = 0.0;
	if (x <= -2.5e-16)
		tmp = t_0;
	elseif (x <= 1.55e-121)
		tmp = t_1;
	elseif (x <= 8.5e-61)
		tmp = t_0;
	elseif (x <= 5.6e-20)
		tmp = t_1;
	elseif ((x <= 1.75e+99) || ~((x <= 2.8e+186)))
		tmp = t_0;
	else
		tmp = abs((x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.5e-16], t$95$0, If[LessEqual[x, 1.55e-121], t$95$1, If[LessEqual[x, 8.5e-61], t$95$0, If[LessEqual[x, 5.6e-20], t$95$1, If[Or[LessEqual[x, 1.75e+99], N[Not[LessEqual[x, 2.8e+186]], $MachinePrecision]], t$95$0, N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000002e-16 or 1.5499999999999999e-121 < x < 8.50000000000000016e-61 or 5.6000000000000005e-20 < x < 1.7499999999999999e99 or 2.80000000000000018e186 < x

   1. Initial program 87.7%

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

   2. Taylor expanded in z around inf 60.2%

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

      1. mul-1-neg 60.2%

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

      2. associate-*l/ 71.8%

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

      3. distribute-rgt-neg-out 71.8%

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

   4. Simplified 71.8%

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

      1. expm1-log1p-u 41.0%

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

      2. expm1-udef 39.4%

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

      3. add-sqr-sqrt 19.8%

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

      4. sqrt-unprod 24.1%

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

      5. sqr-neg 24.1%

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

      6. sqrt-unprod 15.2%

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

      7. add-sqr-sqrt 30.8%

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

      8. associate-*l/ 27.8%

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

      9. *-commutative 27.8%

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

      10. associate-/l* 30.8%

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

   6. Applied egg-rr 30.8%

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

      1. expm1-def 32.5%

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

      2. expm1-log1p 71.8%

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

   8. Simplified 71.8%

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

      1. clear-num 71.8%

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

      2. associate-/r/ 71.7%

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

      3. clear-num 71.8%

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

   10. Applied egg-rr 71.8%

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

   if -2.5000000000000002e-16 < x < 1.5499999999999999e-121 or 8.50000000000000016e-61 < x < 5.6000000000000005e-20

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 83.8%

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

   if 1.7499999999999999e99 < x < 2.80000000000000018e186

   1. Initial program 94.9%

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

   2. Taylor expanded in y around 0 95.1%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0 81.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-16}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+99} \lor \neg \left(x \leq 2.8 \cdot 10^{+186}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+98} \lor \neg \left(x \leq 6.8 \cdot 10^{+187}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* x (/ z y_m)))))
   (if (<= x -4.1e+123)
     t_0
     (if (<= x -3.8e-18)
       t_1
       (if (<= x 2.5e-121)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 8e+98) (not (<= x 6.8e+187))) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -4.1e+123) {
		tmp = t_0;
	} else if (x <= -3.8e-18) {
		tmp = t_1;
	} else if (x <= 2.5e-121) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 8e+98) || !(x <= 6.8e+187)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((x * (z / y_m)))
    if (x <= (-4.1d+123)) then
        tmp = t_0
    else if (x <= (-3.8d-18)) then
        tmp = t_1
    else if (x <= 2.5d-121) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 8d+98) .or. (.not. (x <= 6.8d+187))) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -4.1e+123) {
		tmp = t_0;
	} else if (x <= -3.8e-18) {
		tmp = t_1;
	} else if (x <= 2.5e-121) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 8e+98) || !(x <= 6.8e+187)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -4.1e+123:
		tmp = t_0
	elif x <= -3.8e-18:
		tmp = t_1
	elif x <= 2.5e-121:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 8e+98) or not (x <= 6.8e+187):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -4.1e+123)
		tmp = t_0;
	elseif (x <= -3.8e-18)
		tmp = t_1;
	elseif (x <= 2.5e-121)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 8e+98) || !(x <= 6.8e+187))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -4.1e+123)
		tmp = t_0;
	elseif (x <= -3.8e-18)
		tmp = t_1;
	elseif (x <= 2.5e-121)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 8e+98) || ~((x <= 6.8e+187)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.1e+123], t$95$0, If[LessEqual[x, -3.8e-18], t$95$1, If[LessEqual[x, 2.5e-121], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 8e+98], N[Not[LessEqual[x, 6.8e+187]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.09999999999999989e123 or 7.99999999999999998e98 < x < 6.7999999999999999e187

   1. Initial program 87.2%

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

   2. Taylor expanded in y around 0 91.4%

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

   3. Taylor expanded in x around inf 91.4%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0 74.4%

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

   if -4.09999999999999989e123 < x < -3.7999999999999998e-18 or 2.49999999999999995e-121 < x < 7.99999999999999998e98 or 6.7999999999999999e187 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in z around inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. associate-*l/ 71.0%

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

      3. distribute-rgt-neg-out 71.0%

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

   4. Simplified 71.0%

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

      1. expm1-log1p-u 37.4%

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

      2. expm1-udef 35.2%

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

      3. add-sqr-sqrt 18.4%

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

      4. sqrt-unprod 27.0%

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

      5. sqr-neg 27.0%

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

      6. sqrt-unprod 18.5%

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

      7. add-sqr-sqrt 34.5%

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

      8. associate-*l/ 31.6%

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

      9. *-commutative 31.6%

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

      10. associate-/l* 34.5%

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

   6. Applied egg-rr 34.5%

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

      1. expm1-def 36.6%

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

      2. expm1-log1p 70.9%

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

      3. associate-/r/ 69.1%

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

      4. *-commutative 69.1%

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

   8. Simplified 69.1%

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

   if -3.7999999999999998e-18 < x < 2.49999999999999995e-121

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 84.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+123}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-18}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+98} \lor \neg \left(x \leq 6.8 \cdot 10^{+187}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{x \cdot z}{y_m}\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))))
   (if (<= x -4.2e-15)
     t_0
     (if (<= x 2.4e-121)
       (fabs (/ 4.0 y_m))
       (if (<= x 8e+98)
         (fabs (/ (* x z) y_m))
         (if (<= x 1.55e+184) (fabs (/ x y_m)) t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -4.2e-15) {
		tmp = t_0;
	} else if (x <= 2.4e-121) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 8e+98) {
		tmp = fabs(((x * z) / y_m));
	} else if (x <= 1.55e+184) {
		tmp = fabs((x / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    if (x <= (-4.2d-15)) then
        tmp = t_0
    else if (x <= 2.4d-121) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 8d+98) then
        tmp = abs(((x * z) / y_m))
    else if (x <= 1.55d+184) then
        tmp = abs((x / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -4.2e-15) {
		tmp = t_0;
	} else if (x <= 2.4e-121) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 8e+98) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (x <= 1.55e+184) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -4.2e-15:
		tmp = t_0
	elif x <= 2.4e-121:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 8e+98:
		tmp = math.fabs(((x * z) / y_m))
	elif x <= 1.55e+184:
		tmp = math.fabs((x / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -4.2e-15)
		tmp = t_0;
	elseif (x <= 2.4e-121)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 8e+98)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (x <= 1.55e+184)
		tmp = abs(Float64(x / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -4.2e-15)
		tmp = t_0;
	elseif (x <= 2.4e-121)
		tmp = abs((4.0 / y_m));
	elseif (x <= 8e+98)
		tmp = abs(((x * z) / y_m));
	elseif (x <= 1.55e+184)
		tmp = abs((x / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.2e-15], t$95$0, If[LessEqual[x, 2.4e-121], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 8e+98], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.55e+184], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.19999999999999962e-15 or 1.5499999999999999e184 < x

   1. Initial program 85.3%

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

   2. Taylor expanded in z around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. associate-*l/ 69.9%

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

      3. distribute-rgt-neg-out 69.9%

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

   4. Simplified 69.9%

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

      1. expm1-log1p-u 39.7%

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

      2. expm1-udef 39.5%

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

      3. add-sqr-sqrt 17.1%

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

      4. sqrt-unprod 22.8%

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

      5. sqr-neg 22.8%

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

      6. sqrt-unprod 14.1%

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

      7. add-sqr-sqrt 28.8%

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

      8. associate-*l/ 25.7%

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

      9. *-commutative 25.7%

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

      10. associate-/l* 28.8%

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

   6. Applied egg-rr 28.8%

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

      1. expm1-def 28.9%

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

      2. expm1-log1p 69.8%

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

   8. Simplified 69.8%

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

      1. clear-num 69.8%

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

      2. associate-/r/ 69.8%

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

      3. clear-num 69.9%

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

   10. Applied egg-rr 69.9%

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

   if -4.19999999999999962e-15 < x < 2.40000000000000003e-121

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 84.4%

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

   if 2.40000000000000003e-121 < x < 7.99999999999999998e98

   1. Initial program 92.9%

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

   2. Taylor expanded in z around inf 67.7%

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

      1. mul-1-neg 67.7%

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

      2. associate-*l/ 65.5%

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

      3. distribute-rgt-neg-out 65.5%

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

   4. Simplified 65.5%

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

      1. add-sqr-sqrt 37.1%

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

      2. sqrt-unprod 35.0%

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

      3. sqr-neg 35.0%

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

      4. sqrt-unprod 28.1%

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

      5. add-sqr-sqrt 65.5%

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

      6. associate-*l/ 67.7%

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

   6. Applied egg-rr 67.7%

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

   if 7.99999999999999998e98 < x < 1.5499999999999999e184

   1. Initial program 94.9%

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

   2. Taylor expanded in y around 0 95.1%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0 81.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-15}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{elif}\;x \leq 600000000:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.3e+72)
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (if (<= x 600000000.0)
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
     (fabs (/ x (/ y_m (- 1.0 z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.3e+72) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 600000000.0) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((x / (y_m / (1.0 - z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.3d+72)) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else if (x <= 600000000.0d0) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((x / (y_m / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.3e+72) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 600000000.0) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / (1.0 - z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.3e+72:
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	elif x <= 600000000.0:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((x / (y_m / (1.0 - z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.3e+72)
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	elseif (x <= 600000000.0)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.3e+72)
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	elseif (x <= 600000000.0)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((x / (y_m / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.3e+72], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] + N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 600000000.0], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999991e72

   1. Initial program 83.3%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x around inf 99.9%

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

   if -1.29999999999999991e72 < x < 6e8

   1. Initial program 96.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   if 6e8 < x

   1. Initial program 88.9%

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

   2. Taylor expanded in y around 0 92.6%

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

   3. Taylor expanded in x around inf 92.6%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{elif}\;x \leq 600000000:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 7: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-17} \lor \neg \left(x \leq 2.5 \cdot 10^{-121}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -3.6e-17) (not (<= x 2.5e-121)))
   (fabs (/ x (/ y_m (- 1.0 z))))
   (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -3.6e-17) || !(x <= 2.5e-121)) {
		tmp = fabs((x / (y_m / (1.0 - z))));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.6d-17)) .or. (.not. (x <= 2.5d-121))) then
        tmp = abs((x / (y_m / (1.0d0 - z))))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -3.6e-17) || !(x <= 2.5e-121)) {
		tmp = Math.abs((x / (y_m / (1.0 - z))));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -3.6e-17) or not (x <= 2.5e-121):
		tmp = math.fabs((x / (y_m / (1.0 - z))))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -3.6e-17) || !(x <= 2.5e-121))
		tmp = abs(Float64(x / Float64(y_m / Float64(1.0 - z))));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -3.6e-17) || ~((x <= 2.5e-121)))
		tmp = abs((x / (y_m / (1.0 - z))));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -3.6e-17], N[Not[LessEqual[x, 2.5e-121]], $MachinePrecision]], N[Abs[N[(x / N[(y$95$m / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999995e-17 or 2.49999999999999995e-121 < x

   1. Initial program 88.7%

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

   2. Taylor expanded in y around 0 93.0%

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

   3. Taylor expanded in x around inf 86.4%

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

      1. associate-/l* 92.0%

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

   5. Simplified 92.0%

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

   if -3.59999999999999995e-17 < x < 2.49999999999999995e-121

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 84.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-17} \lor \neg \left(x \leq 2.5 \cdot 10^{-121}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 8: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -2e-17)
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (if (<= x 2.5e-121) (fabs (/ 4.0 y_m)) (fabs (/ x (/ y_m (- 1.0 z)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2e-17) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 2.5e-121) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((x / (y_m / (1.0 - z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2d-17)) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else if (x <= 2.5d-121) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((x / (y_m / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -2e-17) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 2.5e-121) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((x / (y_m / (1.0 - z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -2e-17:
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	elif x <= 2.5e-121:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((x / (y_m / (1.0 - z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -2e-17)
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	elseif (x <= 2.5e-121)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -2e-17)
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	elseif (x <= 2.5e-121)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((x / (y_m / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -2e-17], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] + N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.5e-121], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000014e-17

   1. Initial program 86.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around inf 98.0%

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

   if -2.00000000000000014e-17 < x < 2.49999999999999995e-121

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 84.4%

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

   if 2.49999999999999995e-121 < x

   1. Initial program 90.0%

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

   2. Taylor expanded in y around 0 94.7%

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

   3. Taylor expanded in x around inf 84.8%

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

      1. associate-/l* 88.0%

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

   5. Simplified 88.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-121}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]

Alternative 9: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -8500:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq 4000000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -8500.0)
   (fabs (* z (/ x y_m)))
   (if (<= z 4000000000000.0)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -8500.0) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 4000000000000.0) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x * (z / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8500.0d0)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 4000000000000.0d0) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x * (z / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -8500.0) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 4000000000000.0) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -8500.0:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 4000000000000.0:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -8500.0)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 4000000000000.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -8500.0)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 4000000000000.0)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * (z / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -8500.0], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4000000000000.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8500

   1. Initial program 97.2%

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

   2. Taylor expanded in z around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-*l/ 72.1%

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

      3. distribute-rgt-neg-out 72.1%

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

   4. Simplified 72.1%

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

      1. expm1-log1p-u 39.8%

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

      2. expm1-udef 37.1%

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

      3. add-sqr-sqrt 37.1%

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

      4. sqrt-unprod 26.4%

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

      5. sqr-neg 26.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 33.4%

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

      8. associate-*l/ 32.1%

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

      9. *-commutative 32.1%

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

      10. associate-/l* 33.4%

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

   6. Applied egg-rr 33.4%

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

      1. expm1-def 36.1%

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

      2. expm1-log1p 72.0%

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

   8. Simplified 72.0%

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

      1. clear-num 72.0%

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

      2. associate-/r/ 71.9%

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

      3. clear-num 72.1%

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

   10. Applied egg-rr 72.1%

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

   if -8500 < z < 4e12

   1. Initial program 96.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 97.7%

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

      1. associate-*r/ 97.7%

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

      2. distribute-lft-in 97.7%

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

      3. metadata-eval 97.7%

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

      4. neg-mul-1 97.7%

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

      5. sub-neg 97.7%

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

   6. Simplified 97.7%

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

   if 4e12 < z

   1. Initial program 77.3%

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

   2. Taylor expanded in z around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. associate-*l/ 77.7%

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

      3. distribute-rgt-neg-out 77.7%

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

   4. Simplified 77.7%

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

      1. expm1-log1p-u 46.0%

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

      2. expm1-udef 42.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 22.7%

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

      5. sqr-neg 22.7%

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

      6. sqrt-unprod 31.8%

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

      7. add-sqr-sqrt 31.8%

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

      8. associate-*l/ 30.4%

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

      9. *-commutative 30.4%

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

      10. associate-/l* 31.8%

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

   6. Applied egg-rr 31.8%

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

      1. expm1-def 35.3%

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

      2. expm1-log1p 77.6%

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

      3. associate-/r/ 81.0%

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

      4. *-commutative 81.0%

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

   8. Simplified 81.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8500:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 4000000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 10: 68.7% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 87.7%

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

   2. Taylor expanded in y around 0 91.5%

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

   3. Taylor expanded in x around inf 90.4%

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

      1. associate-/l* 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in z around 0 57.7%

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

   if -10.5 < x < 4

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0 74.6%

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

4. Final simplification 66.5%

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

Alternative 11: 39.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in x around 0 41.1%

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

3. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))