fabs fraction 1

Percentage Accurate: 91.9% → 99.1%

Time: 11.3s

Alternatives: 17

Speedup: 0.9×

• 20.9% 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: 91.9% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \left|t\_0 - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t\_1 \leq 10^{-210} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0 - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)) (t_1 (fabs (- t_0 (* z (/ x y))))))
   (if (or (<= t_1 1e-210) (not (<= t_1 2e+301)))
     (fabs (* (/ -1.0 y) (fma x z (- -4.0 x))))
     (fabs (- t_0 (/ z (/ y x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = fabs((t_0 - (z * (x / y))));
	double tmp;
	if ((t_1 <= 1e-210) || !(t_1 <= 2e+301)) {
		tmp = fabs(((-1.0 / y) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs((t_0 - (z / (y / x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	t_1 = abs(Float64(t_0 - Float64(z * Float64(x / y))))
	tmp = 0.0
	if ((t_1 <= 1e-210) || !(t_1 <= 2e+301))
		tmp = abs(Float64(Float64(-1.0 / y) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(Float64(t_0 - Float64(z / Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(t$95$0 - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$1, 1e-210], N[Not[LessEqual[t$95$1, 2e+301]], $MachinePrecision]], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1e-210 or 2.00000000000000011e301 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 64.8%

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

   3. Simplified 100.0%

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

   if 1e-210 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2.00000000000000011e301

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 10^{-210} \lor \neg \left(\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;\left|t\_0 - z \cdot \frac{x}{y}\right| \leq \infty:\\ \;\;\;\;\left|t\_0 - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \left(x \cdot z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)))
   (if (<= (fabs (- t_0 (* z (/ x y)))) INFINITY)
     (fabs (- t_0 (/ z (/ y x))))
     (fabs (* (/ -1.0 y) (* x z))))))

C

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

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double tmp;
	if (Math.abs((t_0 - (z * (x / y)))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs((t_0 - (z / (y / x))));
	} else {
		tmp = Math.abs(((-1.0 / y) * (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + 4.0) / y
	tmp = 0
	if math.fabs((t_0 - (z * (x / y)))) <= math.inf:
		tmp = math.fabs((t_0 - (z / (y / x))))
	else:
		tmp = math.fabs(((-1.0 / y) * (x * z)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + 4.0) / y;
	tmp = 0.0;
	if (abs((t_0 - (z * (x / y)))) <= Inf)
		tmp = abs((t_0 - (z / (y / x))));
	else
		tmp = abs(((-1.0 / y) * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$0 - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], Infinity], N[Abs[N[(t$95$0 - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

      2. clear-num 97.4%

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

      3. un-div-inv 97.5%

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

   4. Applied egg-rr 97.5%

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

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

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 80.4%

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

4. Final simplification 96.2%

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

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+86)
   (fabs (* (/ -1.0 y) (fma x z (- -4.0 x))))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+86) {
		tmp = fabs(((-1.0 / y) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+86)
		tmp = abs(Float64(Float64(-1.0 / y) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+86], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e86

   1. Initial program 90.1%

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

   3. Simplified 96.1%

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

   if 4.9999999999999998e86 < y

   1. Initial program 90.7%

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

      1. fabs-sub 90.7%

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

      2. associate-*l/ 87.1%

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

      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 4: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \left(x \cdot z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y) (* z (/ x y)))))
   (if (<= t_0 INFINITY) (fabs t_0) (fabs (* (/ -1.0 y) (* x z))))))

C

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

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs(((-1.0 / y) * (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x + 4.0) / y) - (z * (x / y))
	tmp = 0
	if t_0 <= math.inf:
		tmp = math.fabs(t_0)
	else:
		tmp = math.fabs(((-1.0 / y) * (x * z)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x + 4.0) / y) - (z * (x / y));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = abs(t_0);
	else
		tmp = abs(((-1.0 / y) * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z), $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)) < +inf.0

   1. Initial program 97.5%

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 80.4%

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

4. Final simplification 96.2%

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

Alternative 5: 43.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -17000000:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -17000000.0)
   (/ x (/ y (+ -1.0 z)))
   (if (<= x -9.8e-48)
     (* x (/ z (- y)))
     (if (<= x 3.4e+21) (/ (+ x 4.0) y) (* x (/ (- 1.0 z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -17000000.0) {
		tmp = x / (y / (-1.0 + z));
	} else if (x <= -9.8e-48) {
		tmp = x * (z / -y);
	} else if (x <= 3.4e+21) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-17000000.0d0)) then
        tmp = x / (y / ((-1.0d0) + z))
    else if (x <= (-9.8d-48)) then
        tmp = x * (z / -y)
    else if (x <= 3.4d+21) then
        tmp = (x + 4.0d0) / y
    else
        tmp = x * ((1.0d0 - z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -17000000.0) {
		tmp = x / (y / (-1.0 + z));
	} else if (x <= -9.8e-48) {
		tmp = x * (z / -y);
	} else if (x <= 3.4e+21) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -17000000.0:
		tmp = x / (y / (-1.0 + z))
	elif x <= -9.8e-48:
		tmp = x * (z / -y)
	elif x <= 3.4e+21:
		tmp = (x + 4.0) / y
	else:
		tmp = x * ((1.0 - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -17000000.0)
		tmp = Float64(x / Float64(y / Float64(-1.0 + z)));
	elseif (x <= -9.8e-48)
		tmp = Float64(x * Float64(z / Float64(-y)));
	elseif (x <= 3.4e+21)
		tmp = Float64(Float64(x + 4.0) / y);
	else
		tmp = Float64(x * Float64(Float64(1.0 - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -17000000.0)
		tmp = x / (y / (-1.0 + z));
	elseif (x <= -9.8e-48)
		tmp = x * (z / -y);
	elseif (x <= 3.4e+21)
		tmp = (x + 4.0) / y;
	else
		tmp = x * ((1.0 - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -17000000.0], N[(x / N[(y / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.8e-48], N[(x * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e+21], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.7e7

   1. Initial program 80.5%

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

      1. fabs-sub 80.5%

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

      2. associate-*l/ 71.0%

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

      3. associate-*r/ 86.5%

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

      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 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. fma-undefine 49.4%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 46.4%

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

      7. div-inv 46.4%

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

      8. sub-neg 46.4%

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

      9. metadata-eval 46.4%

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

      10. distribute-neg-in 46.4%

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

      11. +-commutative 46.4%

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

      12. cancel-sign-sub-inv 46.4%

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

      13. div-inv 46.4%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 48.4%

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

   6. Applied egg-rr 48.4%

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

   7. Taylor expanded in x around inf 48.4%

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

      1. associate-/l* 56.8%

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

      2. sub-neg 56.8%

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

      3. metadata-eval 56.8%

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

   9. Simplified 56.8%

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

       1. clear-num 56.8%

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

       2. un-div-inv 56.8%

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

   11. Applied egg-rr 56.8%

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

   if -1.7e7 < x < -9.8000000000000005e-48

   1. Initial program 99.4%

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

      1. fabs-sub 99.4%

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

      2. associate-*l/ 99.7%

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

      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. fma-undefine 99.9%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l/ 99.4%

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

      4. div-inv 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

      7. distribute-neg-in 99.3%

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

      8. +-commutative 99.3%

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

      9. cancel-sign-sub-inv 99.3%

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

      10. div-inv 99.4%

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

      11. fabs-sub 99.4%

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

      12. add-sqr-sqrt 63.5%

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

      13. fabs-sqr 63.5%

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

      14. add-sqr-sqrt 64.0%

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

      15. associate-*l/ 64.2%

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

      16. sub-div 64.2%

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

   6. Applied egg-rr 64.2%

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

   7. Taylor expanded in z around inf 47.9%

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

      1. mul-1-neg 47.9%

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

      2. associate-*r/ 48.0%

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

      3. distribute-rgt-neg-in 48.0%

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

      4. distribute-frac-neg2 48.0%

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

   9. Simplified 48.0%

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

   if -9.8000000000000005e-48 < x < 3.4e21

   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/ 99.8%

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

      3. associate-*r/ 86.5%

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

      4. fmm-def 86.5%

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

      5. distribute-neg-frac 86.5%

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

      6. +-commutative 86.5%

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

      7. distribute-neg-in 86.5%

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

      8. unsub-neg 86.5%

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

      9. metadata-eval 86.5%

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

   3. Simplified 86.5%

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

      1. fma-undefine 86.5%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l/ 95.6%

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

      4. div-inv 95.6%

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

      5. sub-neg 95.6%

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

      6. metadata-eval 95.6%

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

      7. distribute-neg-in 95.6%

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

      8. +-commutative 95.6%

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

      9. cancel-sign-sub-inv 95.6%

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

      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 51.0%

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

      13. fabs-sqr 51.0%

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

      14. add-sqr-sqrt 52.0%

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

      15. associate-*l/ 54.1%

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

      16. sub-div 54.1%

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

   6. Applied egg-rr 54.1%

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

   7. Taylor expanded in z around 0 38.8%

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

   if 3.4e21 < x

   1. Initial program 87.1%

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

      1. fabs-sub 87.1%

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

      2. associate-*l/ 79.2%

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

      3. associate-*r/ 87.1%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. fma-undefine 87.1%

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

      2. associate-*r/ 79.2%

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

      3. associate-*l/ 87.1%

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

      4. div-inv 86.9%

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

      5. sub-neg 86.9%

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

      6. metadata-eval 86.9%

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

      7. distribute-neg-in 86.9%

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

      8. +-commutative 86.9%

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

      9. cancel-sign-sub-inv 86.9%

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

      10. div-inv 87.1%

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

      11. fabs-sub 87.1%

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

      12. add-sqr-sqrt 54.9%

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

      13. fabs-sqr 54.9%

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

      14. add-sqr-sqrt 55.5%

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

      15. sub-neg 55.5%

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

      16. distribute-rgt-neg-in 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in x around -inf 63.8%

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

      1. mul-1-neg 63.8%

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

      2. div-sub 63.8%

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

      3. associate-/l* 60.1%

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

      4. sub-neg 60.1%

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

      5. metadata-eval 60.1%

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

      6. distribute-rgt-in 60.1%

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

      7. neg-mul-1 60.1%

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

      8. sub-neg 60.1%

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

      9. *-lft-identity 60.1%

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

      10. sub-neg 60.1%

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

      11. distribute-rgt-in 60.1%

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

      12. *-rgt-identity 60.1%

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

      13. remove-double-neg 60.1%

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

      14. *-commutative 60.1%

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

      15. mul-1-neg 60.1%

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

      16. associate-*r* 60.1%

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

      17. neg-mul-1 60.1%

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

      18. distribute-rgt-neg-in 60.1%

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

      19. mul-1-neg 60.1%

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

      20. distribute-lft-in 60.1%

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

      21. neg-mul-1 60.1%

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

      22. +-commutative 60.1%

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

      23. associate-*r* 60.1%

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

   9. Simplified 63.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -17000000:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -19000000.0)
   (* x (/ (+ -1.0 z) y))
   (if (<= x -1e-47)
     (* x (/ z (- y)))
     (if (<= x 4.8e+23) (/ (+ x 4.0) y) (* x (/ (- 1.0 z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -19000000.0) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= -1e-47) {
		tmp = x * (z / -y);
	} else if (x <= 4.8e+23) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-19000000.0d0)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= (-1d-47)) then
        tmp = x * (z / -y)
    else if (x <= 4.8d+23) then
        tmp = (x + 4.0d0) / y
    else
        tmp = x * ((1.0d0 - z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -19000000.0) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= -1e-47) {
		tmp = x * (z / -y);
	} else if (x <= 4.8e+23) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -19000000.0:
		tmp = x * ((-1.0 + z) / y)
	elif x <= -1e-47:
		tmp = x * (z / -y)
	elif x <= 4.8e+23:
		tmp = (x + 4.0) / y
	else:
		tmp = x * ((1.0 - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -19000000.0)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= -1e-47)
		tmp = Float64(x * Float64(z / Float64(-y)));
	elseif (x <= 4.8e+23)
		tmp = Float64(Float64(x + 4.0) / y);
	else
		tmp = Float64(x * Float64(Float64(1.0 - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -19000000.0)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= -1e-47)
		tmp = x * (z / -y);
	elseif (x <= 4.8e+23)
		tmp = (x + 4.0) / y;
	else
		tmp = x * ((1.0 - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -19000000.0], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-47], N[(x * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e+23], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9e7

   1. Initial program 80.5%

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

      1. fabs-sub 80.5%

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

      2. associate-*l/ 71.0%

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

      3. associate-*r/ 86.5%

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

      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 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. fma-undefine 49.4%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 46.4%

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

      7. div-inv 46.4%

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

      8. sub-neg 46.4%

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

      9. metadata-eval 46.4%

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

      10. distribute-neg-in 46.4%

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

      11. +-commutative 46.4%

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

      12. cancel-sign-sub-inv 46.4%

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

      13. div-inv 46.4%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 48.4%

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

   6. Applied egg-rr 48.4%

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

   7. Taylor expanded in x around inf 48.4%

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

      1. associate-/l* 56.8%

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

      2. sub-neg 56.8%

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

      3. metadata-eval 56.8%

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

   9. Simplified 56.8%

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

   if -1.9e7 < x < -9.9999999999999997e-48

   1. Initial program 99.4%

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

      1. fabs-sub 99.4%

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

      2. associate-*l/ 99.7%

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

      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. fma-undefine 99.9%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l/ 99.4%

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

      4. div-inv 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

      7. distribute-neg-in 99.3%

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

      8. +-commutative 99.3%

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

      9. cancel-sign-sub-inv 99.3%

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

      10. div-inv 99.4%

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

      11. fabs-sub 99.4%

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

      12. add-sqr-sqrt 63.5%

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

      13. fabs-sqr 63.5%

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

      14. add-sqr-sqrt 64.0%

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

      15. associate-*l/ 64.2%

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

      16. sub-div 64.2%

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

   6. Applied egg-rr 64.2%

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

   7. Taylor expanded in z around inf 47.9%

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

      1. mul-1-neg 47.9%

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

      2. associate-*r/ 48.0%

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

      3. distribute-rgt-neg-in 48.0%

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

      4. distribute-frac-neg2 48.0%

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

   9. Simplified 48.0%

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

   if -9.9999999999999997e-48 < x < 4.8e23

   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/ 99.8%

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

      3. associate-*r/ 86.5%

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

      4. fmm-def 86.5%

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

      5. distribute-neg-frac 86.5%

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

      6. +-commutative 86.5%

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

      7. distribute-neg-in 86.5%

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

      8. unsub-neg 86.5%

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

      9. metadata-eval 86.5%

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

   3. Simplified 86.5%

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

      1. fma-undefine 86.5%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l/ 95.6%

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

      4. div-inv 95.6%

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

      5. sub-neg 95.6%

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

      6. metadata-eval 95.6%

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

      7. distribute-neg-in 95.6%

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

      8. +-commutative 95.6%

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

      9. cancel-sign-sub-inv 95.6%

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

      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 51.0%

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

      13. fabs-sqr 51.0%

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

      14. add-sqr-sqrt 52.0%

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

      15. associate-*l/ 54.1%

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

      16. sub-div 54.1%

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

   6. Applied egg-rr 54.1%

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

   7. Taylor expanded in z around 0 38.8%

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

   if 4.8e23 < x

   1. Initial program 87.1%

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

      1. fabs-sub 87.1%

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

      2. associate-*l/ 79.2%

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

      3. associate-*r/ 87.1%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. fma-undefine 87.1%

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

      2. associate-*r/ 79.2%

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

      3. associate-*l/ 87.1%

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

      4. div-inv 86.9%

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

      5. sub-neg 86.9%

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

      6. metadata-eval 86.9%

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

      7. distribute-neg-in 86.9%

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

      8. +-commutative 86.9%

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

      9. cancel-sign-sub-inv 86.9%

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

      10. div-inv 87.1%

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

      11. fabs-sub 87.1%

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

      12. add-sqr-sqrt 54.9%

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

      13. fabs-sqr 54.9%

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

      14. add-sqr-sqrt 55.5%

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

      15. sub-neg 55.5%

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

      16. distribute-rgt-neg-in 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in x around -inf 63.8%

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

      1. mul-1-neg 63.8%

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

      2. div-sub 63.8%

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

      3. associate-/l* 60.1%

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

      4. sub-neg 60.1%

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

      5. metadata-eval 60.1%

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

      6. distribute-rgt-in 60.1%

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

      7. neg-mul-1 60.1%

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

      8. sub-neg 60.1%

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

      9. *-lft-identity 60.1%

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

      10. sub-neg 60.1%

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

      11. distribute-rgt-in 60.1%

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

      12. *-rgt-identity 60.1%

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

      13. remove-double-neg 60.1%

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

      14. *-commutative 60.1%

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

      15. mul-1-neg 60.1%

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

      16. associate-*r* 60.1%

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

      17. neg-mul-1 60.1%

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

      18. distribute-rgt-neg-in 60.1%

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

      19. mul-1-neg 60.1%

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

      20. distribute-lft-in 60.1%

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

      21. neg-mul-1 60.1%

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

      22. +-commutative 60.1%

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

      23. associate-*r* 60.1%

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

   9. Simplified 63.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420000000:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -420000000.0)
   (/ x (/ y (+ -1.0 z)))
   (if (<= x 4.2) (/ (- 4.0 (* x z)) y) (* x (/ (- 1.0 z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -420000000.0) {
		tmp = x / (y / (-1.0 + z));
	} else if (x <= 4.2) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-420000000.0d0)) then
        tmp = x / (y / ((-1.0d0) + z))
    else if (x <= 4.2d0) then
        tmp = (4.0d0 - (x * z)) / y
    else
        tmp = x * ((1.0d0 - z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -420000000.0) {
		tmp = x / (y / (-1.0 + z));
	} else if (x <= 4.2) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -420000000.0:
		tmp = x / (y / (-1.0 + z))
	elif x <= 4.2:
		tmp = (4.0 - (x * z)) / y
	else:
		tmp = x * ((1.0 - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -420000000.0)
		tmp = Float64(x / Float64(y / Float64(-1.0 + z)));
	elseif (x <= 4.2)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y);
	else
		tmp = Float64(x * Float64(Float64(1.0 - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -420000000.0)
		tmp = x / (y / (-1.0 + z));
	elseif (x <= 4.2)
		tmp = (4.0 - (x * z)) / y;
	else
		tmp = x * ((1.0 - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -420000000.0], N[(x / N[(y / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2e8

   1. Initial program 80.5%

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

      1. fabs-sub 80.5%

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

      2. associate-*l/ 71.0%

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

      3. associate-*r/ 86.5%

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

      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 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. fma-undefine 49.4%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 46.4%

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

      7. div-inv 46.4%

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

      8. sub-neg 46.4%

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

      9. metadata-eval 46.4%

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

      10. distribute-neg-in 46.4%

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

      11. +-commutative 46.4%

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

      12. cancel-sign-sub-inv 46.4%

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

      13. div-inv 46.4%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 48.4%

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

   6. Applied egg-rr 48.4%

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

   7. Taylor expanded in x around inf 48.4%

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

      1. associate-/l* 56.8%

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

      2. sub-neg 56.8%

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

      3. metadata-eval 56.8%

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

   9. Simplified 56.8%

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

       1. clear-num 56.8%

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

       2. un-div-inv 56.8%

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

   11. Applied egg-rr 56.8%

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

   if -4.2e8 < x < 4.20000000000000018

   1. Initial program 95.7%

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

      1. fabs-sub 95.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 87.1%

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

      4. fmm-def 87.1%

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

      5. distribute-neg-frac 87.1%

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

      6. +-commutative 87.1%

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

      7. distribute-neg-in 87.1%

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

      8. unsub-neg 87.1%

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

      9. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

      1. fma-undefine 87.1%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l/ 95.7%

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

      4. div-inv 95.7%

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

      5. sub-neg 95.7%

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

      6. metadata-eval 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. +-commutative 95.7%

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

      9. cancel-sign-sub-inv 95.7%

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

      10. div-inv 95.7%

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

      11. fabs-sub 95.7%

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

      12. add-sqr-sqrt 51.0%

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

      13. fabs-sqr 51.0%

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

      14. add-sqr-sqrt 52.0%

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

      15. associate-*l/ 54.0%

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

      16. sub-div 54.0%

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

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in x around 0 53.0%

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

   if 4.20000000000000018 < x

   1. Initial program 88.3%

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

      1. fabs-sub 88.3%

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

      2. associate-*l/ 81.2%

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

      3. associate-*r/ 88.3%

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

      4. fmm-def 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. +-commutative 94.1%

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

      7. distribute-neg-in 94.1%

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

      8. unsub-neg 94.1%

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

      9. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. fma-undefine 88.3%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 88.3%

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

      4. div-inv 88.1%

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

      5. sub-neg 88.1%

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

      6. metadata-eval 88.1%

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

      7. distribute-neg-in 88.1%

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

      8. +-commutative 88.1%

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

      9. cancel-sign-sub-inv 88.1%

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

      10. div-inv 88.3%

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

      11. fabs-sub 88.3%

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

      12. add-sqr-sqrt 57.2%

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

      13. fabs-sqr 57.2%

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

      14. add-sqr-sqrt 57.9%

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

      15. sub-neg 57.9%

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

      16. distribute-rgt-neg-in 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in x around -inf 64.6%

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

      1. mul-1-neg 64.6%

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

      2. div-sub 64.6%

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

      3. associate-/l* 61.3%

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

      4. sub-neg 61.3%

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

      5. metadata-eval 61.3%

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

      6. distribute-rgt-in 61.3%

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

      7. neg-mul-1 61.3%

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

      8. sub-neg 61.3%

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

      9. *-lft-identity 61.3%

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

      10. sub-neg 61.3%

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

      11. distribute-rgt-in 61.3%

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

      12. *-rgt-identity 61.3%

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

      13. remove-double-neg 61.3%

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

      14. *-commutative 61.3%

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

      15. mul-1-neg 61.3%

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

      16. associate-*r* 61.3%

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

      17. neg-mul-1 61.3%

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

      18. distribute-rgt-neg-in 61.3%

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

      19. mul-1-neg 61.3%

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

      20. distribute-lft-in 61.3%

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

      21. neg-mul-1 61.3%

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

      22. +-commutative 61.3%

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

      23. associate-*r* 61.3%

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

   9. Simplified 64.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420000000:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.4% accurate, 6.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0)
   (- (/ (- z) (/ y x)) (/ (+ x 4.0) y))
   (/ (- (+ x 4.0) (* x z)) y)))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-z / (y / x)) - ((x + 4.0) / y);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = (-z / (y / x)) - ((x + 4.0) / y)
	else:
		tmp = ((x + 4.0) - (x * z)) / y
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 81.0%

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

      1. fabs-sub 81.0%

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

      2. associate-*l/ 71.9%

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

      3. associate-*r/ 86.9%

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

      4. fmm-def 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. +-commutative 94.1%

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

      7. distribute-neg-in 94.1%

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

      8. unsub-neg 94.1%

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

      9. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 53.4%

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

      2. fabs-sqr 53.4%

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

      3. add-sqr-sqrt 53.8%

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

      4. fma-undefine 48.0%

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

      5. associate-*r/ 39.7%

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

      6. associate-*l/ 45.1%

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

      7. div-inv 45.1%

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

      8. sub-neg 45.1%

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

      9. metadata-eval 45.1%

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

      10. distribute-neg-in 45.1%

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

      11. +-commutative 45.1%

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

      12. cancel-sign-sub-inv 45.1%

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

      13. div-inv 45.1%

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

      14. associate-*l/ 39.7%

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

      15. associate-*r/ 48.0%

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

   6. Applied egg-rr 48.0%

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

      1. *-commutative 48.0%

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

      2. associate-/r/ 45.1%

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

      3. frac-2neg 45.1%

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

      4. add-sqr-sqrt 23.2%

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

      5. sqrt-unprod 48.3%

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

      6. sqr-neg 48.3%

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

      7. sqrt-unprod 33.4%

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

      8. add-sqr-sqrt 46.8%

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

   8. Applied egg-rr 46.8%

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

   if -4 < x

   1. Initial program 93.6%

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

      1. fabs-sub 93.6%

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

      2. associate-*l/ 94.7%

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

      3. associate-*r/ 87.3%

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

      4. fmm-def 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. +-commutative 88.9%

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

      7. distribute-neg-in 88.9%

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

      8. unsub-neg 88.9%

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

      9. metadata-eval 88.9%

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

   3. Simplified 88.9%

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

      1. fma-undefine 87.3%

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

      2. associate-*r/ 94.7%

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

      3. associate-*l/ 93.6%

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

      4. div-inv 93.6%

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

      5. sub-neg 93.6%

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

      6. metadata-eval 93.6%

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

      7. distribute-neg-in 93.6%

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

      8. +-commutative 93.6%

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

      9. cancel-sign-sub-inv 93.6%

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

      10. div-inv 93.6%

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

      11. fabs-sub 93.6%

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

      12. add-sqr-sqrt 52.2%

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

      13. fabs-sqr 52.2%

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

      14. add-sqr-sqrt 53.1%

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

      15. associate-*l/ 53.6%

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

      16. sub-div 55.7%

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

   6. Applied egg-rr 55.7%

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

4. Final simplification 53.3%

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

Alternative 9: 39.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -380000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -380000000.0)
   (* x (/ (+ -1.0 z) y))
   (if (<= x -1e-47) (* x (/ z (- y))) (/ (+ x 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -380000000.0) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= -1e-47) {
		tmp = x * (z / -y);
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-380000000.0d0)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= (-1d-47)) then
        tmp = x * (z / -y)
    else
        tmp = (x + 4.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -380000000.0) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= -1e-47) {
		tmp = x * (z / -y);
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -380000000.0:
		tmp = x * ((-1.0 + z) / y)
	elif x <= -1e-47:
		tmp = x * (z / -y)
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -380000000.0)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= -1e-47)
		tmp = Float64(x * Float64(z / Float64(-y)));
	else
		tmp = Float64(Float64(x + 4.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -380000000.0)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= -1e-47)
		tmp = x * (z / -y);
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -380000000.0], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-47], N[(x * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e8

   1. Initial program 80.5%

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

      1. fabs-sub 80.5%

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

      2. associate-*l/ 71.0%

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

      3. associate-*r/ 86.5%

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

      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 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. fma-undefine 49.4%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 46.4%

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

      7. div-inv 46.4%

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

      8. sub-neg 46.4%

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

      9. metadata-eval 46.4%

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

      10. distribute-neg-in 46.4%

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

      11. +-commutative 46.4%

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

      12. cancel-sign-sub-inv 46.4%

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

      13. div-inv 46.4%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 48.4%

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

   6. Applied egg-rr 48.4%

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

   7. Taylor expanded in x around inf 48.4%

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

      1. associate-/l* 56.8%

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

      2. sub-neg 56.8%

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

      3. metadata-eval 56.8%

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

   9. Simplified 56.8%

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

   if -3.8e8 < x < -9.9999999999999997e-48

   1. Initial program 99.4%

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

      1. fabs-sub 99.4%

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

      2. associate-*l/ 99.7%

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

      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. fma-undefine 99.9%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l/ 99.4%

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

      4. div-inv 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

      7. distribute-neg-in 99.3%

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

      8. +-commutative 99.3%

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

      9. cancel-sign-sub-inv 99.3%

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

      10. div-inv 99.4%

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

      11. fabs-sub 99.4%

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

      12. add-sqr-sqrt 63.5%

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

      13. fabs-sqr 63.5%

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

      14. add-sqr-sqrt 64.0%

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

      15. associate-*l/ 64.2%

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

      16. sub-div 64.2%

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

   6. Applied egg-rr 64.2%

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

   7. Taylor expanded in z around inf 47.9%

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

      1. mul-1-neg 47.9%

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

      2. associate-*r/ 48.0%

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

      3. distribute-rgt-neg-in 48.0%

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

      4. distribute-frac-neg2 48.0%

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

   9. Simplified 48.0%

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

   if -9.9999999999999997e-48 < x

   1. Initial program 93.3%

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

      1. fabs-sub 93.3%

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

      2. associate-*l/ 94.4%

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

      3. associate-*r/ 86.6%

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

      4. fmm-def 88.3%

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

      5. distribute-neg-frac 88.3%

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

      6. +-commutative 88.3%

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

      7. distribute-neg-in 88.3%

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

      8. unsub-neg 88.3%

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

      9. metadata-eval 88.3%

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

   3. Simplified 88.3%

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

      1. fma-undefine 86.6%

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

      2. associate-*r/ 94.4%

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

      3. associate-*l/ 93.3%

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

      4. div-inv 93.3%

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

      5. sub-neg 93.3%

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

      6. metadata-eval 93.3%

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

      7. distribute-neg-in 93.3%

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

      8. +-commutative 93.3%

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

      9. cancel-sign-sub-inv 93.3%

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

      10. div-inv 93.3%

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

      11. fabs-sub 93.3%

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

      12. add-sqr-sqrt 52.0%

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

      13. fabs-sqr 52.0%

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

      14. add-sqr-sqrt 52.9%

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

      15. associate-*l/ 53.4%

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

      16. sub-div 55.7%

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

   6. Applied egg-rr 55.7%

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

   7. Taylor expanded in z around 0 37.1%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380000000:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.3% accurate, 7.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0) (- (/ (- z) (/ y x)) (/ x y)) (/ (- (+ x 4.0) (* x z)) y)))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-z / (y / x)) - (x / y);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = (-z / (y / x)) - (x / y)
	else:
		tmp = ((x + 4.0) - (x * z)) / y
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 81.0%

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

      1. fabs-sub 81.0%

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

      2. associate-*l/ 71.9%

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

      3. associate-*r/ 86.9%

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

      4. fmm-def 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. +-commutative 94.1%

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

      7. distribute-neg-in 94.1%

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

      8. unsub-neg 94.1%

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

      9. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 53.4%

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

      2. fabs-sqr 53.4%

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

      3. add-sqr-sqrt 53.8%

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

      4. fma-undefine 48.0%

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

      5. associate-*r/ 39.7%

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

      6. associate-*l/ 45.1%

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

      7. div-inv 45.1%

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

      8. sub-neg 45.1%

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

      9. metadata-eval 45.1%

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

      10. distribute-neg-in 45.1%

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

      11. +-commutative 45.1%

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

      12. cancel-sign-sub-inv 45.1%

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

      13. div-inv 45.1%

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

      14. associate-*l/ 39.7%

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

      15. associate-*r/ 48.0%

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

   6. Applied egg-rr 48.0%

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

      1. *-commutative 48.0%

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

      2. associate-/r/ 45.1%

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

      3. frac-2neg 45.1%

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

      4. add-sqr-sqrt 23.2%

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

      5. sqrt-unprod 48.3%

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

      6. sqr-neg 48.3%

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

      7. sqrt-unprod 33.4%

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

      8. add-sqr-sqrt 46.8%

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

   8. Applied egg-rr 46.8%

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

   9. Taylor expanded in x around inf 46.8%

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

   if -4 < x

   1. Initial program 93.6%

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

      1. fabs-sub 93.6%

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

      2. associate-*l/ 94.7%

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

      3. associate-*r/ 87.3%

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

      4. fmm-def 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. +-commutative 88.9%

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

      7. distribute-neg-in 88.9%

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

      8. unsub-neg 88.9%

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

      9. metadata-eval 88.9%

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

   3. Simplified 88.9%

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

      1. fma-undefine 87.3%

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

      2. associate-*r/ 94.7%

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

      3. associate-*l/ 93.6%

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

      4. div-inv 93.6%

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

      5. sub-neg 93.6%

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

      6. metadata-eval 93.6%

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

      7. distribute-neg-in 93.6%

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

      8. +-commutative 93.6%

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

      9. cancel-sign-sub-inv 93.6%

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

      10. div-inv 93.6%

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

      11. fabs-sub 93.6%

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

      12. add-sqr-sqrt 52.2%

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

      13. fabs-sqr 52.2%

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

      14. add-sqr-sqrt 53.1%

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

      15. associate-*l/ 53.6%

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

      16. sub-div 55.7%

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

   6. Applied egg-rr 55.7%

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

4. Final simplification 53.3%

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

Alternative 11: 49.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -31500000:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -31500000.0) (/ x (/ y (+ -1.0 z))) (/ (- (+ x 4.0) (* x z)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -31500000.0) {
		tmp = x / (y / (-1.0 + z));
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-31500000.0d0)) then
        tmp = x / (y / ((-1.0d0) + z))
    else
        tmp = ((x + 4.0d0) - (x * z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -31500000.0) {
		tmp = x / (y / (-1.0 + z));
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -31500000.0:
		tmp = x / (y / (-1.0 + z))
	else:
		tmp = ((x + 4.0) - (x * z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -31500000.0)
		tmp = Float64(x / Float64(y / Float64(-1.0 + z)));
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -31500000.0)
		tmp = x / (y / (-1.0 + z));
	else
		tmp = ((x + 4.0) - (x * z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -31500000.0], N[(x / N[(y / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.15e7

   1. Initial program 80.5%

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

      1. fabs-sub 80.5%

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

      2. associate-*l/ 71.0%

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

      3. associate-*r/ 86.5%

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

      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 54.9%

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

      2. fabs-sqr 54.9%

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

      3. add-sqr-sqrt 55.4%

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

      4. fma-undefine 49.4%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 46.4%

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

      7. div-inv 46.4%

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

      8. sub-neg 46.4%

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

      9. metadata-eval 46.4%

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

      10. distribute-neg-in 46.4%

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

      11. +-commutative 46.4%

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

      12. cancel-sign-sub-inv 46.4%

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

      13. div-inv 46.4%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 48.4%

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

   6. Applied egg-rr 48.4%

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

   7. Taylor expanded in x around inf 48.4%

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

      1. associate-/l* 56.8%

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

      2. sub-neg 56.8%

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

      3. metadata-eval 56.8%

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

   9. Simplified 56.8%

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

       1. clear-num 56.8%

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

       2. un-div-inv 56.8%

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

   11. Applied egg-rr 56.8%

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

   if -3.15e7 < x

   1. Initial program 93.7%

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

      1. fabs-sub 93.7%

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

      2. associate-*l/ 94.7%

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

      3. associate-*r/ 87.4%

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

      4. fmm-def 89.0%

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

      5. distribute-neg-frac 89.0%

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

      6. +-commutative 89.0%

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

      7. distribute-neg-in 89.0%

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

      8. unsub-neg 89.0%

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

      9. metadata-eval 89.0%

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

   3. Simplified 89.0%

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

      1. fma-undefine 87.4%

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

      2. associate-*r/ 94.7%

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

      3. associate-*l/ 93.7%

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

      4. div-inv 93.6%

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

      5. sub-neg 93.6%

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

      6. metadata-eval 93.6%

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

      7. distribute-neg-in 93.6%

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

      8. +-commutative 93.6%

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

      9. cancel-sign-sub-inv 93.6%

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

      10. div-inv 93.7%

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

      11. fabs-sub 93.7%

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

      12. add-sqr-sqrt 52.7%

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

      13. fabs-sqr 52.7%

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

      14. add-sqr-sqrt 53.6%

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

      15. associate-*l/ 54.1%

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

      16. sub-div 56.2%

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

   6. Applied egg-rr 56.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31500000:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55) (/ (- x) y) (if (<= x 4.0) (/ 4.0 y) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55) {
		tmp = -x / y;
	} else if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = -x / y
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55) {
		tmp = -x / y;
	} else if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55:
		tmp = -x / y
	elif x <= 4.0:
		tmp = 4.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(-x) / y);
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = -x / y;
	elseif (x <= 4.0)
		tmp = 4.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55], N[((-x) / y), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 81.0%

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

      1. fabs-sub 81.0%

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

      2. associate-*l/ 71.9%

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

      3. associate-*r/ 86.9%

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

      4. fmm-def 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. +-commutative 94.1%

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

      7. distribute-neg-in 94.1%

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

      8. unsub-neg 94.1%

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

      9. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 53.4%

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

      2. fabs-sqr 53.4%

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

      3. add-sqr-sqrt 53.8%

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

      4. fma-undefine 48.0%

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

      5. associate-*r/ 39.7%

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

      6. associate-*l/ 45.1%

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

      7. div-inv 45.1%

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

      8. sub-neg 45.1%

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

      9. metadata-eval 45.1%

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

      10. distribute-neg-in 45.1%

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

      11. +-commutative 45.1%

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

      12. cancel-sign-sub-inv 45.1%

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

      13. div-inv 45.1%

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

      14. associate-*l/ 39.7%

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

      15. sub-div 47.0%

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

   6. Applied egg-rr 47.0%

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

   7. Taylor expanded in x around inf 47.0%

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

      1. associate-/l* 55.2%

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

      2. sub-neg 55.2%

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

      3. metadata-eval 55.2%

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

   9. Simplified 55.2%

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

   10. Taylor expanded in z around 0 35.1%

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

       1. mul-1-neg 35.1%

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

       2. distribute-frac-neg 35.1%

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

   12. Simplified 35.1%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 95.7%

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

      1. fabs-sub 95.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 86.9%

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

      4. fmm-def 86.9%

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

      5. distribute-neg-frac 86.9%

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

      6. +-commutative 86.9%

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

      7. distribute-neg-in 86.9%

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

      8. unsub-neg 86.9%

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

      9. metadata-eval 86.9%

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

   3. Simplified 86.9%

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

      1. fma-undefine 86.9%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l/ 95.7%

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

      4. div-inv 95.7%

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

      5. sub-neg 95.7%

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

      6. metadata-eval 95.7%

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

      7. distribute-neg-in 95.7%

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

      8. +-commutative 95.7%

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

      9. cancel-sign-sub-inv 95.7%

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

      10. div-inv 95.7%

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

      11. fabs-sub 95.7%

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

      12. add-sqr-sqrt 50.2%

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

      13. fabs-sqr 50.2%

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

      14. add-sqr-sqrt 51.3%

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

      15. associate-*l/ 53.3%

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

      16. sub-div 53.3%

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

   6. Applied egg-rr 53.3%

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

   7. Taylor expanded in x around 0 36.0%

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

   if 4 < x

   1. Initial program 88.3%

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

      1. fabs-sub 88.3%

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

      2. associate-*l/ 81.2%

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

      3. associate-*r/ 88.3%

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

      4. fmm-def 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. +-commutative 94.1%

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

      7. distribute-neg-in 94.1%

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

      8. unsub-neg 94.1%

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

      9. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. fma-undefine 88.3%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 88.3%

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

      4. div-inv 88.1%

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

      5. sub-neg 88.1%

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

      6. metadata-eval 88.1%

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

      7. distribute-neg-in 88.1%

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

      8. +-commutative 88.1%

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

      9. cancel-sign-sub-inv 88.1%

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

      10. div-inv 88.3%

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

      11. fabs-sub 88.3%

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

      12. add-sqr-sqrt 57.2%

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

      13. fabs-sqr 57.2%

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

      14. add-sqr-sqrt 57.9%

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

      15. associate-*l/ 54.3%

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

      16. sub-div 62.0%

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

   6. Applied egg-rr 62.0%

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

   7. Taylor expanded in x around inf 61.3%

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

   8. Taylor expanded in z around 0 34.5%

      \[\leadsto \frac{\color{blue}{x}}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0) (/ (- -4.0 x) y) (/ (+ x 4.0) y)))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-4.0 - x) / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = (-4.0 - x) / y
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 81.0%

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

      1. fabs-sub 81.0%

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

      2. associate-*l/ 71.9%

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

      3. associate-*r/ 86.9%

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

      4. fmm-def 94.1%

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

      5. distribute-neg-frac 94.1%

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

      6. +-commutative 94.1%

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

      7. distribute-neg-in 94.1%

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

      8. unsub-neg 94.1%

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

      9. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 53.4%

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

      2. fabs-sqr 53.4%

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

      3. add-sqr-sqrt 53.8%

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

      4. fma-undefine 48.0%

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

      5. associate-*r/ 39.7%

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

      6. associate-*l/ 45.1%

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

      7. div-inv 45.1%

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

      8. sub-neg 45.1%

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

      9. metadata-eval 45.1%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 45.1%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 45.1%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 45.1%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 45.1%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 39.7%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 47.0%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 35.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 35.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 35.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 35.1%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 35.1%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 35.1%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 35.1%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   if -4 < x

   1. Initial program 93.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 93.6%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 94.7%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.3%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 88.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 87.3%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 94.7%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 93.6%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 93.6%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 93.6%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 93.6%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 52.2%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 52.2%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 53.1%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 53.6%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 55.7%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 36.0%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0) (/ (- x) y) (/ (+ x 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = -x / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.0d0)) then
        tmp = -x / y
    else
        tmp = (x + 4.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = -x / y;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = -x / y
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-x) / y);
	else
		tmp = Float64(Float64(x + 4.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.0)
		tmp = -x / y;
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.0], N[((-x) / y), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 81.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 81.0%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 71.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 86.9%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 94.1%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 94.1%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 53.4%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 53.4%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 53.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 48.0%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 39.7%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 45.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 45.1%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 45.1%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 45.1%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 45.1%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 45.1%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 45.1%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 45.1%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 39.7%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 47.0%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in x around inf 47.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   8. Step-by-step derivation

      1. associate-/l* 55.2%

         \[\leadsto \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 55.2%

         \[\leadsto x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

      3. metadata-eval 55.2%

         \[\leadsto x \cdot \frac{z + \color{blue}{-1}}{y} \]

   9. Simplified 55.2%

      \[\leadsto \color{blue}{x \cdot \frac{z + -1}{y}} \]

   10. Taylor expanded in z around 0 35.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   11. Step-by-step derivation

       1. mul-1-neg 35.1%

          \[\leadsto \color{blue}{-\frac{x}{y}} \]

       2. distribute-frac-neg 35.1%

          \[\leadsto \color{blue}{\frac{-x}{y}} \]

   12. Simplified 35.1%

       \[\leadsto \color{blue}{\frac{-x}{y}} \]

   if -4 < x

   1. Initial program 93.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 93.6%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 94.7%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.3%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 88.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 88.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 87.3%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 94.7%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 93.6%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 93.6%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 93.6%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 93.6%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 93.6%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 52.2%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 52.2%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 53.1%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 53.6%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 55.7%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 36.0%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 4.0) (/ 4.0 y) (/ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = 4.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.0:
		tmp = 4.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(4.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = 4.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.0], N[(4.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 90.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 90.7%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 90.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 86.9%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 89.3%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 89.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 89.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 89.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 89.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 89.3%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 89.3%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 86.9%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 90.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 90.7%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 90.7%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 90.7%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 90.7%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 90.7%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 90.7%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 90.7%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 90.7%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 90.7%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 45.4%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 45.4%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 46.3%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 46.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 48.3%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around 0 25.0%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 88.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 88.3%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 81.2%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 88.3%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 94.1%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 94.1%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 94.1%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 88.3%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 81.2%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 88.3%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 88.1%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 88.1%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 88.1%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 88.1%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 88.1%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 88.1%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 88.3%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 88.3%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 57.2%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 57.2%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 57.9%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 54.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 62.0%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around inf 61.3%

      \[\leadsto \frac{\color{blue}{x} - x \cdot z}{y} \]

   8. Taylor expanded in z around 0 34.5%

      \[\leadsto \frac{\color{blue}{x}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 20.9% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 4.0 y))

C

double code(double x, double y, double z) {
	return 4.0 / y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 4.0d0 / y
end function

Java

public static double code(double x, double y, double z) {
	return 4.0 / y;
}

Python

def code(x, y, z):
	return 4.0 / y

Julia

function code(x, y, z)
	return Float64(4.0 / y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0 / y;
end

Wolfram

code[x_, y_, z_] := N[(4.0 / y), $MachinePrecision]

Derivation

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/ 88.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 87.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 90.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 90.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 87.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

   2. associate-*r/ 88.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

   3. associate-*l/ 90.2%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

   4. div-inv 90.2%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

   5. sub-neg 90.2%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

   6. metadata-eval 90.2%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

   7. distribute-neg-in 90.2%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

   8. +-commutative 90.2%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

   9. cancel-sign-sub-inv 90.2%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

   10. div-inv 90.2%

       \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

   11. fabs-sub 90.2%

       \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

   12. add-sqr-sqrt 47.8%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   13. fabs-sqr 47.8%

       \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   14. add-sqr-sqrt 48.6%

       \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   15. associate-*l/ 48.0%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   16. sub-div 51.1%

       \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

6. Applied egg-rr 51.1%

   \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

7. Taylor expanded in x around 0 20.6%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Alternative 17: 21.1% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ -4.0 y))

C

double code(double x, double y, double z) {
	return -4.0 / y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-4.0d0) / y
end function

Java

public static double code(double x, double y, double z) {
	return -4.0 / y;
}

Python

def code(x, y, z):
	return -4.0 / y

Julia

function code(x, y, z)
	return Float64(-4.0 / y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -4.0 / y;
end

Wolfram

code[x_, y_, z_] := N[(-4.0 / y), $MachinePrecision]

Derivation

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/ 88.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 87.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 90.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 90.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 90.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 43.8%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   2. fabs-sqr 43.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

   3. add-sqr-sqrt 44.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   4. fma-undefine 41.9%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

   5. associate-*r/ 41.8%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

   6. associate-*l/ 42.9%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

   7. div-inv 42.8%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   8. sub-neg 42.8%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

   9. metadata-eval 42.8%

      \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

   10. distribute-neg-in 42.8%

       \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

   11. +-commutative 42.8%

       \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

   12. cancel-sign-sub-inv 42.8%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

   13. div-inv 42.9%

       \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

   14. associate-*l/ 41.8%

       \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

   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 x around 0 19.8%

   \[\leadsto \color{blue}{\frac{-4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))