fabs fraction 1

Percentage Accurate: 91.6% → 99.6%

Time: 7.5s

Alternatives: 9

Speedup: 1.0×

• 20.5% 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: 91.6% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-59)
   (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 <= 2e-59) {
		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 <= 2e-59)
		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, 2e-59], 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 < 2.0000000000000001e-59

   1. Initial program 90.1%

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

   2. Taylor expanded in y around 0 98.3%

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

   if 2.0000000000000001e-59 < y

   1. Initial program 96.2%

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

   3. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y_m} - z \cdot \frac{x}{y_m}\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y_m} \cdot \left(\frac{4}{x} + \left(1 - z\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= (fabs (- (/ (+ 4.0 x) y_m) (* z (/ x y_m)))) 0.1)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (* (/ x y_m) (+ (/ 4.0 x) (- 1.0 z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (fabs((((4.0 + x) / y_m) - (z * (x / y_m)))) <= 0.1) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(((x / y_m) * ((4.0 / x) + (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 (abs((((4.0d0 + x) / y_m) - (z * (x / y_m)))) <= 0.1d0) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs(((x / y_m) * ((4.0d0 / x) + (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 (Math.abs((((4.0 + x) / y_m) - (z * (x / y_m)))) <= 0.1) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((x / y_m) * ((4.0 / x) + (1.0 - z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if math.fabs((((4.0 + x) / y_m) - (z * (x / y_m)))) <= 0.1:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((x / y_m) * ((4.0 / x) + (1.0 - z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (abs(Float64(Float64(Float64(4.0 + x) / y_m) - Float64(z * Float64(x / y_m)))) <= 0.1)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(x / y_m) * Float64(Float64(4.0 / x) + 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 (abs((((4.0 + x) / y_m) - (z * (x / y_m)))) <= 0.1)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs(((x / y_m) * ((4.0 / x) + (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.1], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(N[(4.0 / x), $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 0.10000000000000001

   1. Initial program 90.3%

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

   2. Taylor expanded in y around 0 99.9%

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

   if 0.10000000000000001 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 92.6%

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

      1. *-commutative 92.6%

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

      2. clear-num 92.6%

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

      3. un-div-inv 93.1%

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

   3. Applied egg-rr 93.1%

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

      1. div-inv 92.6%

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

      2. clear-num 92.6%

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

      3. cancel-sign-sub-inv 92.6%

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

      4. +-commutative 92.6%

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

      5. clear-num 92.6%

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

      6. un-div-inv 93.1%

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

      7. add-sqr-sqrt 46.7%

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

      8. sqrt-unprod 77.5%

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

      9. sqr-neg 77.5%

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

      10. sqrt-unprod 52.4%

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

      11. add-sqr-sqrt 90.6%

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

      12. frac-2neg 90.6%

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

      13. frac-add 79.0%

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

   5. Applied egg-rr 79.4%

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

   6. Taylor expanded in y around -inf 95.7%

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

      1. associate-/l* 79.3%

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

      2. associate-/r/ 99.7%

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

      3. associate-+r+ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

      7. mul-1-neg 99.7%

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

      8. unsub-neg 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(\frac{4}{x} + \left(1 - z\right)\right)\right|\\ \end{array} \]

Alternative 3: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.92 \cdot 10^{-44} \lor \neg \left(x \leq 4.8 \cdot 10^{-32}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.92e-44) (not (<= x 4.8e-32)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.92e-44) || !(x <= 4.8e-32)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs(((-4.0 - 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) :: tmp
    if ((x <= (-1.92d-44)) .or. (.not. (x <= 4.8d-32))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs((((-4.0d0) - 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 tmp;
	if ((x <= -1.92e-44) || !(x <= 4.8e-32)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.92e-44) or not (x <= 4.8e-32):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.92e-44) || !(x <= 4.8e-32))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.92e-44) || ~((x <= 4.8e-32)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.92e-44], N[Not[LessEqual[x, 4.8e-32]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.91999999999999992e-44 or 4.8000000000000003e-32 < x

   1. Initial program 90.4%

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

   2. Taylor expanded in y around 0 94.5%

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

   3. Taylor expanded in x around inf 92.3%

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

      1. associate-/l* 97.6%

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

   5. Simplified 97.6%

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

      1. clear-num 97.4%

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

      2. associate-/r/ 97.4%

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

      3. clear-num 97.6%

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

   7. Applied egg-rr 97.6%

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

   if -1.91999999999999992e-44 < x < 4.8000000000000003e-32

   1. Initial program 93.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in z around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. distribute-lft-in 77.9%

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

      3. metadata-eval 77.9%

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

      4. neg-mul-1 77.9%

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

      5. sub-neg 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.92 \cdot 10^{-44} \lor \neg \left(x \leq 4.8 \cdot 10^{-32}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 4: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{1 - z}}\right|\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.5e-47)
   (fabs (/ x (/ y_m (- 1.0 z))))
   (if (<= x 1.2e-31)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (* x (/ (- 1.0 z) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.5e-47) {
		tmp = fabs((x / (y_m / (1.0 - z))));
	} else if (x <= 1.2e-31) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x * ((1.0 - 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 (x <= (-5.5d-47)) then
        tmp = abs((x / (y_m / (1.0d0 - z))))
    else if (x <= 1.2d-31) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x * ((1.0d0 - 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 (x <= -5.5e-47) {
		tmp = Math.abs((x / (y_m / (1.0 - z))));
	} else if (x <= 1.2e-31) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.5e-47:
		tmp = math.fabs((x / (y_m / (1.0 - z))))
	elif x <= 1.2e-31:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.5e-47)
		tmp = abs(Float64(x / Float64(y_m / Float64(1.0 - z))));
	elseif (x <= 1.2e-31)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -5.5e-47)
		tmp = abs((x / (y_m / (1.0 - z))));
	elseif (x <= 1.2e-31)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -5.5e-47], N[Abs[N[(x / N[(y$95$m / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.2e-31], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5000000000000002e-47

   1. Initial program 90.6%

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

   2. Taylor expanded in y around 0 98.7%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/l* 97.8%

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

   5. Simplified 97.8%

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

   if -5.5000000000000002e-47 < x < 1.2e-31

   1. Initial program 93.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in z around 0 77.9%

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

      1. associate-*r/ 77.9%

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

      2. distribute-lft-in 77.9%

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

      3. metadata-eval 77.9%

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

      4. neg-mul-1 77.9%

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

      5. sub-neg 77.9%

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

   6. Simplified 77.9%

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

   if 1.2e-31 < x

   1. Initial program 90.2%

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

   2. Taylor expanded in y around 0 90.8%

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

   3. Taylor expanded in x around inf 88.5%

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

      1. associate-/l* 97.4%

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

   5. Simplified 97.4%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 97.3%

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

      3. clear-num 97.5%

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

   7. Applied egg-rr 97.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 15500000000000:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 15500000000000.0)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (* x (/ (- 1.0 z) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 15500000000000.0) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((x * ((1.0 - 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 (x <= 15500000000000.0d0) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((x * ((1.0d0 - 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 (x <= 15500000000000.0) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 15500000000000.0:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 15500000000000.0)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 15500000000000.0)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 15500000000000.0], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55e13

   1. Initial program 92.8%

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

   2. Taylor expanded in y around 0 99.5%

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

   if 1.55e13 < x

   1. Initial program 89.4%

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

   2. Taylor expanded in y around 0 90.1%

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

   3. Taylor expanded in x around inf 90.1%

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

      1. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. clear-num 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 15500000000000:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 6: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.5)
   (fabs (/ x y_m))
   (if (<= x 1.6e-35) (fabs (/ 4.0 y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.5) {
		tmp = fabs((x / y_m));
	} else if (x <= 1.6e-35) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = fabs((z * (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) :: tmp
    if (x <= (-1.5d0)) then
        tmp = abs((x / y_m))
    else if (x <= 1.6d-35) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = abs((z * (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 tmp;
	if (x <= -1.5) {
		tmp = Math.abs((x / y_m));
	} else if (x <= 1.6e-35) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.5:
		tmp = math.fabs((x / y_m))
	elif x <= 1.6e-35:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.5)
		tmp = abs(Float64(x / y_m));
	elseif (x <= 1.6e-35)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.5)
		tmp = abs((x / y_m));
	elseif (x <= 1.6e-35)
		tmp = abs((4.0 / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.5], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.6e-35], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 90.2%

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

   2. Taylor expanded in z around 0 65.9%

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

      1. associate-*r/ 65.9%

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

      2. metadata-eval 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in x around inf 65.9%

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

   if -1.5 < x < 1.5999999999999999e-35

   1. Initial program 93.7%

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

   2. Taylor expanded in x around 0 77.3%

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

   if 1.5999999999999999e-35 < x

   1. Initial program 90.5%

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

   2. Taylor expanded in z around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. associate-*l/ 68.3%

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

      3. distribute-rgt-neg-out 68.3%

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

   4. Simplified 68.3%

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

      1. add-sqr-sqrt 32.5%

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

      2. sqrt-unprod 46.3%

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

      3. sqr-neg 46.3%

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

      4. sqrt-unprod 35.7%

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

      5. add-sqr-sqrt 68.3%

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

      6. associate-*l/ 50.7%

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

      7. frac-2neg 50.7%

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

      8. distribute-rgt-neg-in 50.7%

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

      9. add-sqr-sqrt 22.4%

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

      10. sqrt-unprod 42.7%

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

      11. sqr-neg 42.7%

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

      12. sqrt-unprod 28.2%

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

      13. add-sqr-sqrt 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. associate-/l* 58.7%

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

      2. associate-/r/ 68.3%

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

      3. add-sqr-sqrt 32.9%

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

      4. sqrt-unprod 55.5%

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

      5. sqr-neg 55.5%

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

      6. sqrt-unprod 35.3%

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

      7. add-sqr-sqrt 68.3%

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

   8. Applied egg-rr 68.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+17}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -9e+17)
   (fabs (* z (/ x y_m)))
   (if (<= z 2.3e+40) (fabs (/ (- -4.0 x) y_m)) (fabs (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -9e+17) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 2.3e+40) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x / (y_m / 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 (z <= (-9d+17)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 2.3d+40) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x / (y_m / 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 (z <= -9e+17) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 2.3e+40) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -9e+17:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 2.3e+40:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -9e+17)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 2.3e+40)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -9e+17)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 2.3e+40)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -9e+17], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.3e+40], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9e17

   1. Initial program 93.7%

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

   2. Taylor expanded in z around inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. associate-*l/ 76.4%

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

      3. distribute-rgt-neg-out 76.4%

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

   4. Simplified 76.4%

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

      1. add-sqr-sqrt 76.2%

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

      2. sqrt-unprod 52.1%

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

      3. sqr-neg 52.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 76.4%

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

      6. associate-*l/ 74.5%

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

      7. frac-2neg 74.5%

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

      8. distribute-rgt-neg-in 74.5%

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

      9. add-sqr-sqrt 74.2%

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

      10. sqrt-unprod 49.5%

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

      11. sqr-neg 49.5%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. associate-/l* 72.7%

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

      2. associate-/r/ 76.4%

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

      3. add-sqr-sqrt 31.4%

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

      4. sqrt-unprod 60.9%

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

      5. sqr-neg 60.9%

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

      6. sqrt-unprod 44.7%

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

      7. add-sqr-sqrt 76.4%

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

   8. Applied egg-rr 76.4%

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

   if -9e17 < z < 2.29999999999999994e40

   1. Initial program 95.0%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 98.1%

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

      1. associate-*r/ 98.1%

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

      2. distribute-lft-in 98.1%

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

      3. metadata-eval 98.1%

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

      4. neg-mul-1 98.1%

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

      5. sub-neg 98.1%

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

   6. Simplified 98.1%

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

   if 2.29999999999999994e40 < z

   1. Initial program 82.0%

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

   2. Taylor expanded in z around inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-*l/ 76.1%

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

      3. distribute-rgt-neg-out 76.1%

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

   4. Simplified 76.1%

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

      1. associate-*l/ 76.9%

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

      2. associate-/l* 77.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 48.8%

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

      5. sqr-neg 48.8%

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

      6. sqrt-unprod 76.9%

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

      7. add-sqr-sqrt 77.2%

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

   6. Applied egg-rr 77.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+17}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 8: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.55) (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 <= -1.55) || !(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 <= (-1.55d0)) .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 <= -1.55) || !(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 <= -1.55) 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 <= -1.55) || !(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 <= -1.55) || ~((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, -1.55], 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 < -1.55000000000000004 or 4 < x

   1. Initial program 89.8%

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

   2. Taylor expanded in z around 0 62.3%

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

      1. associate-*r/ 62.3%

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

      2. metadata-eval 62.3%

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

   4. Simplified 62.3%

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

   5. Taylor expanded in x around inf 62.3%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0 74.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 9: 40.2% 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 39.5%

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

3. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))