fabs fraction 1

Percentage Accurate: 91.4% → 99.7%

Time: 10.9s

Alternatives: 16

Speedup: 0.9×

• 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: 91.4% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-50)
   (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))
   (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-50) {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	} 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-50)
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	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-50], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $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 < 4.00000000000000003e-50

   1. Initial program 92.0%

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

   3. Simplified 98.7%

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

   if 4.00000000000000003e-50 < y

   1. Initial program 97.7%

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

      1. fabs-sub 97.7%

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

      2. associate-*l/ 92.0%

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

      3. associate-*r/ 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;t\_0 - z \cdot \frac{x}{y\_m} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\left|t\_0 - \frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= (- t_0 (* z (/ x y_m))) -2e-43)
     (fabs (- t_0 (/ z (/ y_m x))))
     (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if ((t_0 - (z * (x / y_m))) <= -2e-43) {
		tmp = fabs((t_0 - (z / (y_m / x))));
	} else {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (Float64(t_0 - Float64(z * Float64(x / y_m))) <= -2e-43)
		tmp = abs(Float64(t_0 - Float64(z / Float64(y_m / x))));
	else
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-43], N[Abs[N[(t$95$0 - N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -2.00000000000000015e-43

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -2.00000000000000015e-43 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 90.3%

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

   3. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* z (/ x y_m)))))
   (if (<= t_0 INFINITY) (fabs t_0) (/ x (- y_m)))))

C

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = ((x + 4.0) / y_m) - (z * (x / y_m))
	tmp = 0
	if t_0 <= math.inf:
		tmp = math.fabs(t_0)
	else:
		tmp = x / -y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = abs(t_0);
	else
		tmp = Float64(x / Float64(-y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = ((x + 4.0) / y_m) - (z * (x / y_m));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = abs(t_0);
	else
		tmp = 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[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[Abs[t$95$0], $MachinePrecision], N[(x / (-y$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.1%

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

   if +inf.0 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.0%

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

      1. fabs-sub 0.0%

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

      2. associate-*l/ 18.2%

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

      3. associate-*r/ 18.2%

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

      4. fmm-def 54.5%

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

      5. distribute-neg-frac 54.5%

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

      6. +-commutative 54.5%

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

      7. distribute-neg-in 54.5%

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

      8. unsub-neg 54.5%

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

      9. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

      1. add-sqr-sqrt 27.3%

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

      2. fabs-sqr 27.3%

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

      3. add-sqr-sqrt 27.3%

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

      4. fma-undefine 9.1%

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

      5. associate-*r/ 9.1%

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

      6. associate-*l/ 0.0%

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

      7. div-inv 0.0%

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

      8. sub-neg 0.0%

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

      9. metadata-eval 0.0%

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

      10. distribute-neg-in 0.0%

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

      11. +-commutative 0.0%

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

      12. cancel-sign-sub-inv 0.0%

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

      13. div-inv 0.0%

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

      14. associate-*l/ 9.1%

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

      15. sub-div 27.3%

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

   6. Applied egg-rr 27.3%

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

   7. Taylor expanded in z around 0 54.5%

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

      1. associate-*r/ 54.5%

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

      2. distribute-lft-in 54.5%

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

      3. metadata-eval 54.5%

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

      4. neg-mul-1 54.5%

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

      5. sub-neg 54.5%

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

   9. Simplified 54.5%

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

   10. Taylor expanded in x around inf 54.5%

       \[\leadsto \frac{\color{blue}{-1 \cdot x}}{y} \]
   11. Step-by-step derivation

       1. neg-mul-1 54.5%

          \[\leadsto \frac{\color{blue}{-x}}{y} \]

   12. Simplified 54.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq \infty:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{+252}:\\ \;\;\;\;x \cdot \frac{z}{y\_m} - t\_0\\ \mathbf{elif}\;x \leq -1250000:\\ \;\;\;\;\frac{z}{\frac{y\_m}{-x}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)))
   (if (<= x -9.6e+252)
     (- (* x (/ z y_m)) t_0)
     (if (<= x -1250000.0)
       (- (/ z (/ y_m (- x))) t_0)
       (/ (- (+ x 4.0) (* x z)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -9.6e+252) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= -1250000.0) {
		tmp = (z / (y_m / -x)) - t_0;
	} else {
		tmp = ((x + 4.0) - (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) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    if (x <= (-9.6d+252)) then
        tmp = (x * (z / y_m)) - t_0
    else if (x <= (-1250000.0d0)) then
        tmp = (z / (y_m / -x)) - t_0
    else
        tmp = ((x + 4.0d0) - (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 t_0 = (x + 4.0) / y_m;
	double tmp;
	if (x <= -9.6e+252) {
		tmp = (x * (z / y_m)) - t_0;
	} else if (x <= -1250000.0) {
		tmp = (z / (y_m / -x)) - t_0;
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	tmp = 0
	if x <= -9.6e+252:
		tmp = (x * (z / y_m)) - t_0
	elif x <= -1250000.0:
		tmp = (z / (y_m / -x)) - t_0
	else:
		tmp = ((x + 4.0) - (x * z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	tmp = 0.0
	if (x <= -9.6e+252)
		tmp = Float64(Float64(x * Float64(z / y_m)) - t_0);
	elseif (x <= -1250000.0)
		tmp = Float64(Float64(z / Float64(y_m / Float64(-x))) - t_0);
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	tmp = 0.0;
	if (x <= -9.6e+252)
		tmp = (x * (z / y_m)) - t_0;
	elseif (x <= -1250000.0)
		tmp = (z / (y_m / -x)) - t_0;
	else
		tmp = ((x + 4.0) - (x * z)) / 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[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, If[LessEqual[x, -9.6e+252], N[(N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, -1250000.0], N[(N[(z / N[(y$95$m / (-x)), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5999999999999996e252

   1. Initial program 92.3%

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

      1. fabs-sub 92.3%

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

      2. associate-*l/ 93.2%

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

      3. associate-*r/ 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 45.9%

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

      2. fabs-sqr 45.9%

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

      3. add-sqr-sqrt 46.1%

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

      4. fma-undefine 46.1%

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

      5. associate-*r/ 39.4%

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

      6. associate-*l/ 38.5%

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

      7. div-inv 38.4%

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

      8. sub-neg 38.4%

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

      9. metadata-eval 38.4%

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

      10. distribute-neg-in 38.4%

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

      11. +-commutative 38.4%

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

      12. cancel-sign-sub-inv 38.4%

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

      13. div-inv 38.5%

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

      14. associate-*l/ 39.4%

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

      15. associate-*r/ 46.1%

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

   6. Applied egg-rr 46.1%

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

   if -9.5999999999999996e252 < x < -1.25e6

   1. Initial program 87.4%

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

      1. fabs-sub 87.4%

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

      2. associate-*l/ 83.6%

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

      3. associate-*r/ 89.4%

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

      4. fmm-def 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. +-commutative 93.6%

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

      7. distribute-neg-in 93.6%

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

      8. unsub-neg 93.6%

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

      9. metadata-eval 93.6%

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

   3. Simplified 93.6%

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

      1. add-sqr-sqrt 49.7%

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

      2. fabs-sqr 49.7%

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

      3. add-sqr-sqrt 50.3%

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

      4. fma-undefine 48.2%

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

      5. associate-*r/ 42.4%

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

      6. associate-*l/ 48.2%

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

      7. div-inv 48.1%

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

      8. sub-neg 48.1%

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

      9. metadata-eval 48.1%

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

      10. distribute-neg-in 48.1%

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

      11. +-commutative 48.1%

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

      12. cancel-sign-sub-inv 48.1%

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

      13. div-inv 48.2%

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

      14. associate-*l/ 42.4%

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

      15. associate-*r/ 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. *-commutative 48.2%

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

      2. associate-/r/ 48.2%

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

      3. frac-2neg 48.2%

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

      4. add-sqr-sqrt 20.9%

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

      5. sqrt-unprod 41.0%

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

      6. sqr-neg 41.0%

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

      7. sqrt-unprod 23.8%

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

      8. add-sqr-sqrt 40.6%

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

   8. Applied egg-rr 40.6%

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

   if -1.25e6 < x

   1. Initial program 95.5%

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

      1. fabs-sub 95.5%

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

      2. associate-*l/ 95.4%

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

      3. associate-*r/ 92.6%

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

      4. fmm-def 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. +-commutative 93.6%

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

      7. distribute-neg-in 93.6%

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

      8. unsub-neg 93.6%

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

      9. metadata-eval 93.6%

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

   3. Simplified 93.6%

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

      1. fma-undefine 92.6%

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

      2. associate-*r/ 95.4%

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

      3. associate-*l/ 95.5%

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

      4. div-inv 95.5%

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

      5. sub-neg 95.5%

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

      6. metadata-eval 95.5%

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

      7. distribute-neg-in 95.5%

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

      8. +-commutative 95.5%

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

      9. cancel-sign-sub-inv 95.5%

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

      10. div-inv 95.5%

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

      11. fabs-sub 95.5%

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

      12. add-sqr-sqrt 50.5%

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

      13. fabs-sqr 50.5%

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

      14. add-sqr-sqrt 51.4%

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

      15. associate-*l/ 51.9%

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

      16. sub-div 53.4%

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

   6. Applied egg-rr 53.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+252}:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{elif}\;x \leq -1250000:\\ \;\;\;\;\frac{z}{\frac{y}{-x}} - \frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -26000000000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 0.0003:\\ \;\;\;\;\frac{4 - x \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -26000000000000.0)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 0.0003) (/ (- 4.0 (* x z)) y_m) (* (/ x y_m) (+ z 1.0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -26000000000000.0) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 0.0003) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x / y_m) * (z + 1.0);
	}
	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) :: tmp
    if (x <= (-26000000000000.0d0)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 0.0003d0) then
        tmp = (4.0d0 - (x * z)) / y_m
    else
        tmp = (x / y_m) * (z + 1.0d0)
    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 <= -26000000000000.0) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 0.0003) {
		tmp = (4.0 - (x * z)) / y_m;
	} else {
		tmp = (x / y_m) * (z + 1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -26000000000000.0:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 0.0003:
		tmp = (4.0 - (x * z)) / y_m
	else:
		tmp = (x / y_m) * (z + 1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -26000000000000.0)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 0.0003)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y_m);
	else
		tmp = Float64(Float64(x / y_m) * Float64(z + 1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -26000000000000.0)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 0.0003)
		tmp = (4.0 - (x * z)) / y_m;
	else
		tmp = (x / y_m) * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -26000000000000.0], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0003], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e13

   1. Initial program 88.1%

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

      1. fabs-sub 88.1%

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

      2. associate-*l/ 85.2%

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

      3. associate-*r/ 91.4%

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

      4. fmm-def 94.8%

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

      5. distribute-neg-frac 94.8%

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

      6. +-commutative 94.8%

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

      7. distribute-neg-in 94.8%

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

      8. unsub-neg 94.8%

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

      9. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 48.8%

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

      2. fabs-sqr 48.8%

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

      3. add-sqr-sqrt 49.2%

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

      4. fma-undefine 47.5%

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

      5. associate-*r/ 41.3%

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

      6. associate-*l/ 45.9%

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

      7. div-inv 45.8%

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

      8. sub-neg 45.8%

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

      9. metadata-eval 45.8%

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

      10. distribute-neg-in 45.8%

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

      11. +-commutative 45.8%

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

      12. cancel-sign-sub-inv 45.8%

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

      13. div-inv 45.9%

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

      14. associate-*l/ 41.3%

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

      15. sub-div 43.0%

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

   6. Applied egg-rr 43.0%

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

   7. Taylor expanded in x around inf 43.0%

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

      1. associate-/l* 49.1%

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

      2. sub-neg 49.1%

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

      3. metadata-eval 49.1%

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

   9. Simplified 49.1%

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

   if -2.6e13 < x < 2.99999999999999974e-4

   1. Initial program 96.2%

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

      1. fabs-sub 96.2%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 91.5%

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

      4. fmm-def 91.5%

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

      5. distribute-neg-frac 91.5%

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

      6. +-commutative 91.5%

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

      7. distribute-neg-in 91.5%

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

      8. unsub-neg 91.5%

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

      9. metadata-eval 91.5%

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

   3. Simplified 91.5%

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

      1. fma-undefine 91.5%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 96.2%

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

      4. div-inv 96.1%

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

      5. sub-neg 96.1%

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

      6. metadata-eval 96.1%

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

      7. distribute-neg-in 96.1%

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

      8. +-commutative 96.1%

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

      9. cancel-sign-sub-inv 96.1%

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

      10. div-inv 96.2%

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

      11. fabs-sub 96.2%

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

      12. add-sqr-sqrt 53.8%

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

      13. fabs-sqr 53.8%

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

      14. add-sqr-sqrt 54.9%

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

      15. associate-*l/ 56.4%

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

      16. sub-div 56.4%

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

   6. Applied egg-rr 56.4%

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

   7. Taylor expanded in x around 0 54.1%

      \[\leadsto \frac{\color{blue}{4} - x \cdot z}{y} \]

   if 2.99999999999999974e-4 < x

   1. Initial program 94.6%

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

      1. fabs-sub 94.6%

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

      2. associate-*l/ 88.3%

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

      3. associate-*r/ 94.6%

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

      4. fmm-def 97.3%

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

      5. distribute-neg-frac 97.3%

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

      6. +-commutative 97.3%

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

      7. distribute-neg-in 97.3%

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

      8. unsub-neg 97.3%

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

      9. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. fma-undefine 94.6%

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

      2. associate-*r/ 88.3%

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

      3. associate-*l/ 94.6%

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

      4. div-inv 94.4%

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

      5. sub-neg 94.4%

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

      6. metadata-eval 94.4%

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

      7. distribute-neg-in 94.4%

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

      8. +-commutative 94.4%

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

      9. cancel-sign-sub-inv 94.4%

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

      10. div-inv 94.6%

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

      11. fabs-sub 94.6%

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

      12. add-sqr-sqrt 45.1%

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

      13. fabs-sqr 45.1%

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

      14. add-sqr-sqrt 45.7%

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

      15. sub-neg 45.7%

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

      16. distribute-rgt-neg-in 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in x around inf 45.7%

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

      1. *-un-lft-identity 45.7%

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

      2. *-commutative 45.7%

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

      3. distribute-rgt-out 49.7%

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

      4. add-sqr-sqrt 16.1%

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

      5. sqrt-unprod 41.9%

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

      6. sqr-neg 41.9%

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

      7. sqrt-unprod 29.5%

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

      8. add-sqr-sqrt 55.9%

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

      9. +-commutative 55.9%

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

   9. Applied egg-rr 55.9%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -26000000000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.0003:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.8e-19)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 3.3e-54) (/ 4.0 y_m) (* (/ x y_m) (+ z 1.0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.8e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 3.3e-54) {
		tmp = 4.0 / y_m;
	} else {
		tmp = (x / y_m) * (z + 1.0);
	}
	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) :: tmp
    if (x <= (-5.8d-19)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 3.3d-54) then
        tmp = 4.0d0 / y_m
    else
        tmp = (x / y_m) * (z + 1.0d0)
    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 <= -5.8e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 3.3e-54) {
		tmp = 4.0 / y_m;
	} else {
		tmp = (x / y_m) * (z + 1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.8e-19:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 3.3e-54:
		tmp = 4.0 / y_m
	else:
		tmp = (x / y_m) * (z + 1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.8e-19)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 3.3e-54)
		tmp = Float64(4.0 / y_m);
	else
		tmp = Float64(Float64(x / y_m) * Float64(z + 1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -5.8e-19)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 3.3e-54)
		tmp = 4.0 / y_m;
	else
		tmp = (x / y_m) * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -5.8e-19], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-54], N[(4.0 / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e-19

   1. Initial program 90.2%

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

      1. fabs-sub 90.2%

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

      2. associate-*l/ 87.8%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

      1. add-sqr-sqrt 46.9%

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

      2. fabs-sqr 46.9%

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

      3. add-sqr-sqrt 47.6%

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

      4. fma-undefine 46.2%

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

      5. associate-*r/ 41.1%

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

      6. associate-*l/ 44.8%

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

      7. div-inv 44.8%

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

      8. sub-neg 44.8%

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

      9. metadata-eval 44.8%

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

      10. distribute-neg-in 44.8%

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

      11. +-commutative 44.8%

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

      12. cancel-sign-sub-inv 44.8%

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

      13. div-inv 44.8%

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

      14. associate-*l/ 41.1%

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

      15. sub-div 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in x around inf 41.4%

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

      1. associate-/l* 46.4%

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

      2. sub-neg 46.4%

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

      3. metadata-eval 46.4%

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

   9. Simplified 46.4%

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

   if -5.8e-19 < x < 3.29999999999999993e-54

   1. Initial program 95.4%

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

      1. fabs-sub 95.4%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 89.7%

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

      4. fmm-def 89.7%

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

      5. distribute-neg-frac 89.7%

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

      6. +-commutative 89.7%

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

      7. distribute-neg-in 89.7%

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

      8. unsub-neg 89.7%

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

      9. metadata-eval 89.7%

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

   3. Simplified 89.7%

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

      1. fma-undefine 89.7%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.4%

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

      4. div-inv 95.4%

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

      5. sub-neg 95.4%

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

      6. metadata-eval 95.4%

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

      7. distribute-neg-in 95.4%

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

      8. +-commutative 95.4%

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

      9. cancel-sign-sub-inv 95.4%

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

      10. div-inv 95.4%

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

      11. fabs-sub 95.4%

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

      12. add-sqr-sqrt 51.8%

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

      13. fabs-sqr 51.8%

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

      14. add-sqr-sqrt 53.0%

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

      15. associate-*l/ 54.8%

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

      16. sub-div 54.8%

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

   6. Applied egg-rr 54.8%

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

   7. Taylor expanded in x around 0 47.5%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 3.29999999999999993e-54 < x

   1. Initial program 95.1%

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

      1. fabs-sub 95.1%

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

      2. associate-*l/ 89.6%

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

      3. associate-*r/ 95.1%

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

      4. fmm-def 97.5%

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

      5. distribute-neg-frac 97.5%

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

      6. +-commutative 97.5%

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

      7. distribute-neg-in 97.5%

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

      8. unsub-neg 97.5%

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

      9. metadata-eval 97.5%

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

   3. Simplified 97.5%

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

      1. fma-undefine 95.1%

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

      2. associate-*r/ 89.6%

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

      3. associate-*l/ 95.1%

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

      4. div-inv 95.0%

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

      5. sub-neg 95.0%

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

      6. metadata-eval 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. +-commutative 95.0%

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

      9. cancel-sign-sub-inv 95.0%

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

      10. div-inv 95.1%

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

      11. fabs-sub 95.1%

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

      12. add-sqr-sqrt 47.3%

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

      13. fabs-sqr 47.3%

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

      14. add-sqr-sqrt 47.9%

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

      15. sub-neg 47.9%

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

      16. distribute-rgt-neg-in 47.9%

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

   6. Applied egg-rr 47.9%

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

   7. Taylor expanded in x around inf 45.8%

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

      1. *-un-lft-identity 45.8%

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

      2. *-commutative 45.8%

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

      3. distribute-rgt-out 49.4%

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

      4. add-sqr-sqrt 15.6%

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

      5. sqrt-unprod 38.9%

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

      6. sqr-neg 38.9%

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

      7. sqrt-unprod 27.8%

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

      8. add-sqr-sqrt 51.4%

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

      9. +-commutative 51.4%

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

   9. Applied egg-rr 51.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.3e-19)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 3.5e+14) (/ (+ x 4.0) y_m) (* (/ x y_m) (- 1.0 z)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.3e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 3.5e+14) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = (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 <= (-4.3d-19)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 3.5d+14) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = (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 <= -4.3e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 3.5e+14) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = (x / y_m) * (1.0 - z);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.3e-19:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 3.5e+14:
		tmp = (x + 4.0) / y_m
	else:
		tmp = (x / y_m) * (1.0 - z)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.3e-19)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 3.5e+14)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = Float64(Float64(x / 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 <= -4.3e-19)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 3.5e+14)
		tmp = (x + 4.0) / y_m;
	else
		tmp = (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, -4.3e-19], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+14], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e-19

   1. Initial program 90.2%

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

      1. fabs-sub 90.2%

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

      2. associate-*l/ 87.8%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

      1. add-sqr-sqrt 46.9%

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

      2. fabs-sqr 46.9%

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

      3. add-sqr-sqrt 47.6%

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

      4. fma-undefine 46.2%

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

      5. associate-*r/ 41.1%

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

      6. associate-*l/ 44.8%

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

      7. div-inv 44.8%

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

      8. sub-neg 44.8%

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

      9. metadata-eval 44.8%

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

      10. distribute-neg-in 44.8%

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

      11. +-commutative 44.8%

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

      12. cancel-sign-sub-inv 44.8%

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

      13. div-inv 44.8%

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

      14. associate-*l/ 41.1%

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

      15. sub-div 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in x around inf 41.4%

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

      1. associate-/l* 46.4%

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

      2. sub-neg 46.4%

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

      3. metadata-eval 46.4%

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

   9. Simplified 46.4%

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

   if -4.3e-19 < x < 3.5e14

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 90.8%

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

      4. fmm-def 90.8%

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

      5. distribute-neg-frac 90.8%

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

      6. +-commutative 90.8%

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

      7. distribute-neg-in 90.8%

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

      8. unsub-neg 90.8%

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

      9. metadata-eval 90.8%

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

   3. Simplified 90.8%

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

      1. fma-undefine 90.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.9%

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

      4. div-inv 95.8%

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

      5. sub-neg 95.8%

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

      6. metadata-eval 95.8%

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

      7. distribute-neg-in 95.8%

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

      8. +-commutative 95.8%

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

      9. cancel-sign-sub-inv 95.8%

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

      10. div-inv 95.9%

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

      11. fabs-sub 95.9%

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

      12. add-sqr-sqrt 52.4%

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

      13. fabs-sqr 52.4%

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

      14. add-sqr-sqrt 53.6%

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

      15. associate-*l/ 55.2%

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

      16. sub-div 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in z around 0 44.4%

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

   if 3.5e14 < x

   1. Initial program 94.4%

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

      1. fabs-sub 94.4%

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

      2. associate-*l/ 87.8%

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

      3. associate-*r/ 94.4%

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

      4. fmm-def 97.2%

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

      5. distribute-neg-frac 97.2%

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

      6. +-commutative 97.2%

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

      7. distribute-neg-in 97.2%

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

      8. unsub-neg 97.2%

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

      9. metadata-eval 97.2%

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

   3. Simplified 97.2%

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

      1. fma-undefine 94.4%

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

      2. associate-*r/ 87.8%

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

      3. associate-*l/ 94.4%

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

      4. div-inv 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. +-commutative 94.2%

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

      9. cancel-sign-sub-inv 94.2%

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

      10. div-inv 94.4%

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

      11. fabs-sub 94.4%

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

      12. add-sqr-sqrt 45.6%

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

      13. fabs-sqr 45.6%

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

      14. add-sqr-sqrt 46.2%

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

      15. associate-*l/ 44.9%

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

      16. sub-div 49.0%

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

   6. Applied egg-rr 49.0%

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

   7. Taylor expanded in x around inf 49.0%

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

      1. *-commutative 49.0%

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

      2. associate-/l* 50.3%

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

   9. Simplified 50.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.8e-19)
   (* x (/ (+ -1.0 z) y_m))
   (if (<= x 3.5e+14) (/ (+ x 4.0) y_m) (* x (/ (- 1.0 z) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.8e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 3.5e+14) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = 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 <= (-5.8d-19)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else if (x <= 3.5d+14) then
        tmp = (x + 4.0d0) / y_m
    else
        tmp = 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 <= -5.8e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else if (x <= 3.5e+14) {
		tmp = (x + 4.0) / y_m;
	} else {
		tmp = x * ((1.0 - z) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.8e-19:
		tmp = x * ((-1.0 + z) / y_m)
	elif x <= 3.5e+14:
		tmp = (x + 4.0) / y_m
	else:
		tmp = x * ((1.0 - z) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.8e-19)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	elseif (x <= 3.5e+14)
		tmp = Float64(Float64(x + 4.0) / y_m);
	else
		tmp = 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 <= -5.8e-19)
		tmp = x * ((-1.0 + z) / y_m);
	elseif (x <= 3.5e+14)
		tmp = (x + 4.0) / y_m;
	else
		tmp = 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, -5.8e-19], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+14], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e-19

   1. Initial program 90.2%

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

      1. fabs-sub 90.2%

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

      2. associate-*l/ 87.8%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

      1. add-sqr-sqrt 46.9%

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

      2. fabs-sqr 46.9%

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

      3. add-sqr-sqrt 47.6%

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

      4. fma-undefine 46.2%

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

      5. associate-*r/ 41.1%

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

      6. associate-*l/ 44.8%

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

      7. div-inv 44.8%

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

      8. sub-neg 44.8%

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

      9. metadata-eval 44.8%

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

      10. distribute-neg-in 44.8%

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

      11. +-commutative 44.8%

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

      12. cancel-sign-sub-inv 44.8%

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

      13. div-inv 44.8%

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

      14. associate-*l/ 41.1%

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

      15. sub-div 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in x around inf 41.4%

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

      1. associate-/l* 46.4%

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

      2. sub-neg 46.4%

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

      3. metadata-eval 46.4%

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

   9. Simplified 46.4%

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

   if -5.8e-19 < x < 3.5e14

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 90.8%

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

      4. fmm-def 90.8%

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

      5. distribute-neg-frac 90.8%

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

      6. +-commutative 90.8%

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

      7. distribute-neg-in 90.8%

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

      8. unsub-neg 90.8%

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

      9. metadata-eval 90.8%

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

   3. Simplified 90.8%

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

      1. fma-undefine 90.8%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.9%

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

      4. div-inv 95.8%

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

      5. sub-neg 95.8%

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

      6. metadata-eval 95.8%

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

      7. distribute-neg-in 95.8%

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

      8. +-commutative 95.8%

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

      9. cancel-sign-sub-inv 95.8%

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

      10. div-inv 95.9%

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

      11. fabs-sub 95.9%

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

      12. add-sqr-sqrt 52.4%

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

      13. fabs-sqr 52.4%

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

      14. add-sqr-sqrt 53.6%

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

      15. associate-*l/ 55.2%

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

      16. sub-div 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in z around 0 44.4%

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

   if 3.5e14 < x

   1. Initial program 94.4%

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

      1. fabs-sub 94.4%

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

      2. associate-*l/ 87.8%

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

      3. associate-*r/ 94.4%

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

      4. fmm-def 97.2%

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

      5. distribute-neg-frac 97.2%

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

      6. +-commutative 97.2%

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

      7. distribute-neg-in 97.2%

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

      8. unsub-neg 97.2%

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

      9. metadata-eval 97.2%

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

   3. Simplified 97.2%

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

      1. fma-undefine 94.4%

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

      2. associate-*r/ 87.8%

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

      3. associate-*l/ 94.4%

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

      4. div-inv 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. distribute-neg-in 94.2%

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

      8. +-commutative 94.2%

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

      9. cancel-sign-sub-inv 94.2%

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

      10. div-inv 94.4%

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

      11. fabs-sub 94.4%

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

      12. add-sqr-sqrt 45.6%

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

      13. fabs-sqr 45.6%

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

      14. add-sqr-sqrt 46.2%

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

      15. sub-neg 46.2%

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

      16. distribute-rgt-neg-in 46.2%

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in x around -inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. div-sub 50.3%

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

      3. associate-/l* 49.0%

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

      4. sub-neg 49.0%

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

      5. metadata-eval 49.0%

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

      6. distribute-rgt-in 49.0%

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

      7. neg-mul-1 49.0%

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

      8. sub-neg 49.0%

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

      9. *-lft-identity 49.0%

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

      10. sub-neg 49.0%

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

      11. distribute-rgt-in 49.0%

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

      12. *-rgt-identity 49.0%

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

      13. remove-double-neg 49.0%

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

      14. *-commutative 49.0%

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

      15. mul-1-neg 49.0%

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

      16. associate-*r* 49.0%

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

      17. neg-mul-1 49.0%

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

      18. distribute-rgt-neg-in 49.0%

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

      19. mul-1-neg 49.0%

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

      20. distribute-lft-in 49.0%

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

      21. neg-mul-1 49.0%

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

      22. +-commutative 49.0%

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

      23. associate-*r* 49.0%

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

   9. Simplified 50.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -26000000000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -26000000000000.0)
   (* x (/ (+ -1.0 z) y_m))
   (/ (- (+ x 4.0) (* x z)) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -26000000000000.0) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = ((x + 4.0) - (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 (x <= (-26000000000000.0d0)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else
        tmp = ((x + 4.0d0) - (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 (x <= -26000000000000.0) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -26000000000000.0:
		tmp = x * ((-1.0 + z) / y_m)
	else:
		tmp = ((x + 4.0) - (x * z)) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -26000000000000.0)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * 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 <= -26000000000000.0)
		tmp = x * ((-1.0 + z) / y_m);
	else
		tmp = ((x + 4.0) - (x * z)) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -26000000000000.0], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e13

   1. Initial program 88.1%

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

      1. fabs-sub 88.1%

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

      2. associate-*l/ 85.2%

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

      3. associate-*r/ 91.4%

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

      4. fmm-def 94.8%

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

      5. distribute-neg-frac 94.8%

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

      6. +-commutative 94.8%

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

      7. distribute-neg-in 94.8%

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

      8. unsub-neg 94.8%

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

      9. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 48.8%

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

      2. fabs-sqr 48.8%

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

      3. add-sqr-sqrt 49.2%

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

      4. fma-undefine 47.5%

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

      5. associate-*r/ 41.3%

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

      6. associate-*l/ 45.9%

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

      7. div-inv 45.8%

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

      8. sub-neg 45.8%

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

      9. metadata-eval 45.8%

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

      10. distribute-neg-in 45.8%

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

      11. +-commutative 45.8%

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

      12. cancel-sign-sub-inv 45.8%

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

      13. div-inv 45.9%

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

      14. associate-*l/ 41.3%

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

      15. sub-div 43.0%

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

   6. Applied egg-rr 43.0%

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

   7. Taylor expanded in x around inf 43.0%

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

      1. associate-/l* 49.1%

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

      2. sub-neg 49.1%

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

      3. metadata-eval 49.1%

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

   9. Simplified 49.1%

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

   if -2.6e13 < x

   1. Initial program 95.6%

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

      1. fabs-sub 95.6%

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

      2. associate-*l/ 95.5%

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

      3. associate-*r/ 92.7%

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

      4. fmm-def 93.7%

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

      5. distribute-neg-frac 93.7%

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

      6. +-commutative 93.7%

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

      7. distribute-neg-in 93.7%

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

      8. unsub-neg 93.7%

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

      9. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

      1. fma-undefine 92.7%

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

      2. associate-*r/ 95.5%

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

      3. associate-*l/ 95.6%

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

      4. div-inv 95.5%

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

      5. sub-neg 95.5%

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

      6. metadata-eval 95.5%

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

      7. distribute-neg-in 95.5%

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

      8. +-commutative 95.5%

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

      9. cancel-sign-sub-inv 95.5%

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

      10. div-inv 95.6%

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

      11. fabs-sub 95.6%

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

      12. add-sqr-sqrt 50.5%

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

      13. fabs-sqr 50.5%

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

      14. add-sqr-sqrt 51.4%

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

      15. associate-*l/ 51.9%

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

      16. sub-div 53.4%

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

   6. Applied egg-rr 53.4%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -26000000000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.6% accurate, 8.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -10.5) (/ x (- y_m)) (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -10.5) {
		tmp = x / -y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = 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 <= (-10.5d0)) then
        tmp = x / -y_m
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = 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 <= -10.5) {
		tmp = x / -y_m;
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -10.5:
		tmp = x / -y_m
	elif x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -10.5)
		tmp = Float64(x / Float64(-y_m));
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = 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 <= -10.5)
		tmp = x / -y_m;
	elseif (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -10.5], N[(x / (-y$95$m)), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.5

   1. Initial program 88.9%

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

      1. fabs-sub 88.9%

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

      2. associate-*l/ 86.3%

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

      3. associate-*r/ 92.0%

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

      4. fmm-def 95.2%

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

      5. distribute-neg-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. unsub-neg 95.2%

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

      9. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. add-sqr-sqrt 49.7%

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

      2. fabs-sqr 49.7%

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

      3. add-sqr-sqrt 50.2%

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

      4. fma-undefine 48.6%

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

      5. associate-*r/ 42.9%

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

      6. associate-*l/ 47.1%

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

      7. div-inv 47.0%

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

      8. sub-neg 47.0%

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

      9. metadata-eval 47.0%

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

      10. distribute-neg-in 47.0%

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

      11. +-commutative 47.0%

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

      12. cancel-sign-sub-inv 47.0%

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

      13. div-inv 47.1%

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

      14. associate-*l/ 42.9%

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

      15. sub-div 44.5%

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

   6. Applied egg-rr 44.5%

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

   7. Taylor expanded in z around 0 28.6%

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

      1. associate-*r/ 28.6%

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

      2. distribute-lft-in 28.6%

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

      3. metadata-eval 28.6%

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

      4. neg-mul-1 28.6%

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

      5. sub-neg 28.6%

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

   9. Simplified 28.6%

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

   10. Taylor expanded in x around inf 28.6%

       \[\leadsto \frac{\color{blue}{-1 \cdot x}}{y} \]
   11. Step-by-step derivation

       1. neg-mul-1 28.6%

          \[\leadsto \frac{\color{blue}{-x}}{y} \]

   12. Simplified 28.6%

       \[\leadsto \frac{\color{blue}{-x}}{y} \]

   if -10.5 < x < 4

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 91.1%

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

      4. fmm-def 91.1%

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

      5. distribute-neg-frac 91.1%

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

      6. +-commutative 91.1%

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

      7. distribute-neg-in 91.1%

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

      8. unsub-neg 91.1%

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

      9. metadata-eval 91.1%

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

   3. Simplified 91.1%

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

      1. fma-undefine 91.1%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 96.0%

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

      4. div-inv 96.0%

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

      5. sub-neg 96.0%

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

      6. metadata-eval 96.0%

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

      7. distribute-neg-in 96.0%

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

      8. +-commutative 96.0%

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

      9. cancel-sign-sub-inv 96.0%

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

      10. div-inv 96.0%

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

      11. fabs-sub 96.0%

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

      12. add-sqr-sqrt 54.4%

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

      13. fabs-sqr 54.4%

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

      14. add-sqr-sqrt 55.6%

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

      15. associate-*l/ 57.1%

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

      16. sub-div 57.1%

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

   6. Applied egg-rr 57.1%

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

   7. Taylor expanded in x around 0 42.7%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 94.6%

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

      1. fabs-sub 94.6%

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

      2. associate-*l/ 88.3%

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

      3. associate-*r/ 94.6%

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

      4. fmm-def 97.3%

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

      5. distribute-neg-frac 97.3%

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

      6. +-commutative 97.3%

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

      7. distribute-neg-in 97.3%

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

      8. unsub-neg 97.3%

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

      9. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. fma-undefine 94.6%

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

      2. associate-*r/ 88.3%

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

      3. associate-*l/ 94.6%

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

      4. div-inv 94.4%

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

      5. sub-neg 94.4%

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

      6. metadata-eval 94.4%

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

      7. distribute-neg-in 94.4%

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

      8. +-commutative 94.4%

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

      9. cancel-sign-sub-inv 94.4%

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

      10. div-inv 94.6%

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

      11. fabs-sub 94.6%

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

      12. add-sqr-sqrt 45.1%

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

      13. fabs-sqr 45.1%

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

      14. add-sqr-sqrt 45.7%

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

      15. sub-neg 45.7%

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

      16. distribute-rgt-neg-in 45.7%

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

   6. Applied egg-rr 45.7%

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

   7. Taylor expanded in x around -inf 49.6%

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

      1. mul-1-neg 49.6%

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

      2. div-sub 49.6%

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

      3. associate-/l* 48.4%

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

      4. sub-neg 48.4%

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

      5. metadata-eval 48.4%

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

      6. distribute-rgt-in 48.4%

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

      7. neg-mul-1 48.4%

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

      8. sub-neg 48.4%

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

      9. *-lft-identity 48.4%

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

      10. sub-neg 48.4%

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

      11. distribute-rgt-in 48.4%

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

      12. *-rgt-identity 48.4%

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

      13. remove-double-neg 48.4%

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

      14. *-commutative 48.4%

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

      15. mul-1-neg 48.4%

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

      16. associate-*r* 48.4%

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

      17. neg-mul-1 48.4%

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

      18. distribute-rgt-neg-in 48.4%

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

      19. mul-1-neg 48.4%

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

      20. distribute-lft-in 48.4%

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

      21. neg-mul-1 48.4%

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

      22. +-commutative 48.4%

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

      23. associate-*r* 48.4%

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

   9. Simplified 49.6%

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

   10. Taylor expanded in z around 0 32.9%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.8% accurate, 9.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.5e-19) (* x (/ (+ -1.0 z) y_m)) (/ (+ x 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -5.5e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = (x + 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 <= (-5.5d-19)) then
        tmp = x * (((-1.0d0) + z) / y_m)
    else
        tmp = (x + 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 <= -5.5e-19) {
		tmp = x * ((-1.0 + z) / y_m);
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -5.5e-19:
		tmp = x * ((-1.0 + z) / y_m)
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -5.5e-19)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y_m));
	else
		tmp = Float64(Float64(x + 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 <= -5.5e-19)
		tmp = x * ((-1.0 + z) / y_m);
	else
		tmp = (x + 4.0) / 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-19], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999996e-19

   1. Initial program 90.2%

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

      1. fabs-sub 90.2%

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

      2. associate-*l/ 87.8%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. unsub-neg 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

      1. add-sqr-sqrt 46.9%

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

      2. fabs-sqr 46.9%

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

      3. add-sqr-sqrt 47.6%

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

      4. fma-undefine 46.2%

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

      5. associate-*r/ 41.1%

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

      6. associate-*l/ 44.8%

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

      7. div-inv 44.8%

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

      8. sub-neg 44.8%

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

      9. metadata-eval 44.8%

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

      10. distribute-neg-in 44.8%

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

      11. +-commutative 44.8%

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

      12. cancel-sign-sub-inv 44.8%

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

      13. div-inv 44.8%

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

      14. associate-*l/ 41.1%

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

      15. sub-div 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in x around inf 41.4%

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

      1. associate-/l* 46.4%

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

      2. sub-neg 46.4%

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

      3. metadata-eval 46.4%

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

   9. Simplified 46.4%

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

   if -5.4999999999999996e-19 < x

   1. Initial program 95.3%

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

      1. fabs-sub 95.3%

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

      2. associate-*l/ 95.2%

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

      3. associate-*r/ 92.2%

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

      4. fmm-def 93.3%

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

      5. distribute-neg-frac 93.3%

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

      6. +-commutative 93.3%

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

      7. distribute-neg-in 93.3%

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

      8. unsub-neg 93.3%

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

      9. metadata-eval 93.3%

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

   3. Simplified 93.3%

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

      1. fma-undefine 92.2%

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

      2. associate-*r/ 95.2%

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

      3. associate-*l/ 95.3%

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

      4. div-inv 95.2%

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

      5. sub-neg 95.2%

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

      6. metadata-eval 95.2%

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

      7. distribute-neg-in 95.2%

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

      8. +-commutative 95.2%

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

      9. cancel-sign-sub-inv 95.2%

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

      10. div-inv 95.3%

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

      11. fabs-sub 95.3%

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

      12. add-sqr-sqrt 49.7%

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

      13. fabs-sqr 49.7%

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

      14. add-sqr-sqrt 50.7%

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

      15. associate-*l/ 51.2%

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

      16. sub-div 52.8%

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

   6. Applied egg-rr 52.8%

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

   7. Taylor expanded in z around 0 40.4%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 88.9%

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

      1. fabs-sub 88.9%

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

      2. associate-*l/ 86.3%

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

      3. associate-*r/ 92.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 95.2%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.7%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 49.7%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 48.6%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 42.9%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 47.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 47.0%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 47.0%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 47.0%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 47.0%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 47.0%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 47.0%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 47.1%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 42.9%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 44.5%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 44.5%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 28.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 28.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 28.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 28.6%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 28.6%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 28.6%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 28.6%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   if -4 < x

   1. Initial program 95.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 95.5%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 95.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 93.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.5%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 92.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 95.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 95.5%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 95.4%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 95.5%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 95.5%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 50.8%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 50.8%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 51.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 52.1%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 53.7%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 39.8%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.3% accurate, 11.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x}{-y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4.0) (/ x (- y_m)) (/ (+ x 4.0) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = x / -y_m;
	} else {
		tmp = (x + 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 <= (-4.0d0)) then
        tmp = x / -y_m
    else
        tmp = (x + 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 <= -4.0) {
		tmp = x / -y_m;
	} else {
		tmp = (x + 4.0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4.0:
		tmp = x / -y_m
	else:
		tmp = (x + 4.0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(x / Float64(-y_m));
	else
		tmp = Float64(Float64(x + 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 <= -4.0)
		tmp = x / -y_m;
	else
		tmp = (x + 4.0) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -4.0], N[(x / (-y$95$m)), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 88.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 88.9%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 86.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 95.2%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 95.2%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 49.7%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 49.7%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 48.6%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 42.9%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 47.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 47.0%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 47.0%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 47.0%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 47.0%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 47.0%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 47.0%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 47.1%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 42.9%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 44.5%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 44.5%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 28.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 28.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 28.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 28.6%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 28.6%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 28.6%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 28.6%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   10. Taylor expanded in x around inf 28.6%

       \[\leadsto \frac{\color{blue}{-1 \cdot x}}{y} \]
   11. Step-by-step derivation

       1. neg-mul-1 28.6%

          \[\leadsto \frac{\color{blue}{-x}}{y} \]

   12. Simplified 28.6%

       \[\leadsto \frac{\color{blue}{-x}}{y} \]

   if -4 < x

   1. Initial program 95.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 95.5%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 95.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 93.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.5%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 92.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 95.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 95.5%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 95.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 95.4%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 95.5%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 95.5%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 50.8%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 50.8%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 51.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 52.1%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 53.7%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 39.8%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m)))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = 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 <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = 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 <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = 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 <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 93.5%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 93.5%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 95.1%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 91.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.6%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 91.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 95.1%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 93.5%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 93.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 93.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 93.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 93.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 93.5%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 93.5%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 93.5%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 93.5%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 50.0%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 50.0%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 51.0%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 52.5%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 54.7%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around 0 28.7%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 94.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 94.6%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 88.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 94.6%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 97.3%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 97.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 97.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 97.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 97.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 97.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 94.6%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 88.3%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 94.6%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 94.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 94.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 94.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 94.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 94.4%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 94.4%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 94.6%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 94.6%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 45.1%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 45.1%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 45.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. sub-neg 45.7%

          \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      16. distribute-rgt-neg-in 45.7%

          \[\leadsto \frac{x + 4}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)} \]

   6. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{\frac{x + 4}{y} + \frac{x}{y} \cdot \left(-z\right)} \]

   7. Taylor expanded in x around -inf 49.6%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 49.6%

         \[\leadsto \color{blue}{-x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)} \]

      2. div-sub 49.6%

         \[\leadsto -x \cdot \color{blue}{\frac{z - 1}{y}} \]

      3. associate-/l* 48.4%

         \[\leadsto -\color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]

      4. sub-neg 48.4%

         \[\leadsto -\frac{x \cdot \color{blue}{\left(z + \left(-1\right)\right)}}{y} \]

      5. metadata-eval 48.4%

         \[\leadsto -\frac{x \cdot \left(z + \color{blue}{-1}\right)}{y} \]

      6. distribute-rgt-in 48.4%

         \[\leadsto -\frac{\color{blue}{z \cdot x + -1 \cdot x}}{y} \]

      7. neg-mul-1 48.4%

         \[\leadsto -\frac{z \cdot x + \color{blue}{\left(-x\right)}}{y} \]

      8. sub-neg 48.4%

         \[\leadsto -\frac{\color{blue}{z \cdot x - x}}{y} \]

      9. *-lft-identity 48.4%

         \[\leadsto -\frac{\color{blue}{1 \cdot \left(z \cdot x - x\right)}}{y} \]

      10. sub-neg 48.4%

          \[\leadsto -\frac{1 \cdot \color{blue}{\left(z \cdot x + \left(-x\right)\right)}}{y} \]

      11. distribute-rgt-in 48.4%

          \[\leadsto -\frac{\color{blue}{\left(z \cdot x\right) \cdot 1 + \left(-x\right) \cdot 1}}{y} \]

      12. *-rgt-identity 48.4%

          \[\leadsto -\frac{\color{blue}{z \cdot x} + \left(-x\right) \cdot 1}{y} \]

      13. remove-double-neg 48.4%

          \[\leadsto -\frac{\color{blue}{\left(-\left(-z \cdot x\right)\right)} + \left(-x\right) \cdot 1}{y} \]

      14. *-commutative 48.4%

          \[\leadsto -\frac{\left(-\left(-\color{blue}{x \cdot z}\right)\right) + \left(-x\right) \cdot 1}{y} \]

      15. mul-1-neg 48.4%

          \[\leadsto -\frac{\left(-\color{blue}{-1 \cdot \left(x \cdot z\right)}\right) + \left(-x\right) \cdot 1}{y} \]

      16. associate-*r* 48.4%

          \[\leadsto -\frac{\left(-\color{blue}{\left(-1 \cdot x\right) \cdot z}\right) + \left(-x\right) \cdot 1}{y} \]

      17. neg-mul-1 48.4%

          \[\leadsto -\frac{\left(-\color{blue}{\left(-x\right)} \cdot z\right) + \left(-x\right) \cdot 1}{y} \]

      18. distribute-rgt-neg-in 48.4%

          \[\leadsto -\frac{\color{blue}{\left(-x\right) \cdot \left(-z\right)} + \left(-x\right) \cdot 1}{y} \]

      19. mul-1-neg 48.4%

          \[\leadsto -\frac{\left(-x\right) \cdot \color{blue}{\left(-1 \cdot z\right)} + \left(-x\right) \cdot 1}{y} \]

      20. distribute-lft-in 48.4%

          \[\leadsto -\frac{\color{blue}{\left(-x\right) \cdot \left(-1 \cdot z + 1\right)}}{y} \]

      21. neg-mul-1 48.4%

          \[\leadsto -\frac{\color{blue}{\left(-1 \cdot x\right)} \cdot \left(-1 \cdot z + 1\right)}{y} \]

      22. +-commutative 48.4%

          \[\leadsto -\frac{\left(-1 \cdot x\right) \cdot \color{blue}{\left(1 + -1 \cdot z\right)}}{y} \]

      23. associate-*r* 48.4%

          \[\leadsto -\frac{\color{blue}{-1 \cdot \left(x \cdot \left(1 + -1 \cdot z\right)\right)}}{y} \]

   9. Simplified 49.6%

      \[\leadsto \color{blue}{x \cdot \frac{1 - z}{y}} \]

   10. Taylor expanded in z around 0 32.9%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 40.2% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ 4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return 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 = 4.0d0 / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return 4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return 4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = 4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 93.8%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 93.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 92.4%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 93.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 93.9%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 92.4%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

   2. associate-*r/ 93.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

   3. associate-*l/ 93.8%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

   4. div-inv 93.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

   5. sub-neg 93.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

   6. metadata-eval 93.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

   7. distribute-neg-in 93.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

   8. +-commutative 93.7%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

   9. cancel-sign-sub-inv 93.7%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

   10. div-inv 93.8%

       \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

   11. fabs-sub 93.8%

       \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

   12. add-sqr-sqrt 48.6%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   13. fabs-sqr 48.6%

       \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   14. add-sqr-sqrt 49.4%

       \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   15. associate-*l/ 50.1%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   16. sub-div 52.9%

       \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

6. Applied egg-rr 52.9%

   \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

7. Taylor expanded in x around 0 21.1%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Alternative 16: 1.7% accurate, 37.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{-4}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ -4.0 y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return -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 = (-4.0d0) / y_m
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return -4.0 / y_m;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return -4.0 / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(-4.0 / y_m)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = -4.0 / y_m;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(-4.0 / y$95$m), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 93.8%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 93.1%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 92.4%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 93.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 93.9%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 93.9%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 44.5%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   2. fabs-sqr 44.5%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

   3. add-sqr-sqrt 45.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   4. fma-undefine 44.7%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

   5. associate-*r/ 44.3%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

   6. associate-*l/ 45.8%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

   7. div-inv 45.7%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   8. sub-neg 45.7%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

   9. metadata-eval 45.7%

      \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

   10. distribute-neg-in 45.7%

       \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

   11. +-commutative 45.7%

       \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

   12. cancel-sign-sub-inv 45.7%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

   13. div-inv 45.8%

       \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

   14. associate-*l/ 44.3%

       \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

   15. sub-div 45.1%

       \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

6. Applied egg-rr 45.1%

   \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

7. Taylor expanded in x around 0 16.7%

   \[\leadsto \color{blue}{\frac{-4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))