fabs fraction 1

Percentage Accurate: 91.8% → 98.8%

Time: 9.0s

Alternatives: 15

Speedup: 1.0×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% 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: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -650 \lor \neg \left(x \leq 3.85\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -650.0) (not (<= x 3.85)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- (* x z) 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -650.0) || !(x <= 3.85)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs((((x * z) - 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 <= (-650.0d0)) .or. (.not. (x <= 3.85d0))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs((((x * z) - 4.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -650.0) || !(x <= 3.85)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs((((x * z) - 4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -650.0) or not (x <= 3.85):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs((((x * z) - 4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -650.0) || !(x <= 3.85))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(Float64(Float64(x * z) - 4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -650.0) || ~((x <= 3.85)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs((((x * z) - 4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -650.0], N[Not[LessEqual[x, 3.85]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -650 or 3.85000000000000009 < x

   1. Initial program 84.9%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in x around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. associate-/l* 98.9%

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

      3. associate-*r* 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*r/ 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

      8. distribute-lft-in 98.9%

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

      9. neg-mul-1 98.9%

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

      10. metadata-eval 98.9%

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

      11. +-commutative 98.9%

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

      12. unsub-neg 98.9%

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

   7. Simplified 98.9%

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

   if -650 < x < 3.85000000000000009

   1. Initial program 94.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -650 \lor \neg \left(x \leq 3.85\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+94}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq -400000:\\ \;\;\;\;\frac{x - \left(x \cdot z - 4\right)}{y}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.4e+94)
   (fabs (/ (* x z) y))
   (if (<= z -400000.0)
     (/ (- x (- (* x z) 4.0)) y)
     (if (<= z 1.75e+26) (fabs (/ (+ x 4.0) y)) (fabs (* x (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+94) {
		tmp = fabs(((x * z) / y));
	} else if (z <= -400000.0) {
		tmp = (x - ((x * z) - 4.0)) / y;
	} else if (z <= 1.75e+26) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((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 (z <= (-1.4d+94)) then
        tmp = abs(((x * z) / y))
    else if (z <= (-400000.0d0)) then
        tmp = (x - ((x * z) - 4.0d0)) / y
    else if (z <= 1.75d+26) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+94) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= -400000.0) {
		tmp = (x - ((x * z) - 4.0)) / y;
	} else if (z <= 1.75e+26) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.4e+94:
		tmp = math.fabs(((x * z) / y))
	elif z <= -400000.0:
		tmp = (x - ((x * z) - 4.0)) / y
	elif z <= 1.75e+26:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.4e+94)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= -400000.0)
		tmp = Float64(Float64(x - Float64(Float64(x * z) - 4.0)) / y);
	elseif (z <= 1.75e+26)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.4e+94)
		tmp = abs(((x * z) / y));
	elseif (z <= -400000.0)
		tmp = (x - ((x * z) - 4.0)) / y;
	elseif (z <= 1.75e+26)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.4e+94], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -400000.0], N[(N[(x - N[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1.75e+26], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.39999999999999999e94

   1. Initial program 90.5%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in z around inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. distribute-frac-neg 84.4%

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

      3. distribute-lft-neg-in 84.4%

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

      4. *-commutative 84.4%

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

   7. Simplified 84.4%

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

   if -1.39999999999999999e94 < z < -4e5

   1. Initial program 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. fmm-def 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. rem-square-sqrt 53.6%

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

      10. fabs-sqr 53.6%

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

      11. rem-square-sqrt 55.1%

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

      12. distribute-frac-neg2 55.1%

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

      13. distribute-frac-neg 55.1%

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

      14. neg-sub0 55.1%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 55.1%

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

   8. Taylor expanded in x around 0 55.1%

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

   if -4e5 < z < 1.75e26

   1. Initial program 93.6%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.4%

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

      1. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   if 1.75e26 < z

   1. Initial program 75.6%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in z around inf 76.3%

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

      1. mul-1-neg 76.3%

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

      2. associate-*r/ 77.9%

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

      3. *-commutative 77.9%

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

      4. distribute-rgt-neg-in 77.9%

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

   7. Simplified 77.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+94}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq -400000:\\ \;\;\;\;\frac{x - \left(x \cdot z - 4\right)}{y}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;z \leq -1 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -48000:\\ \;\;\;\;\frac{x + \left(4 - x \cdot z\right)}{y}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y)))))
   (if (<= z -1e+94)
     t_0
     (if (<= z -48000.0)
       (/ (+ x (- 4.0 (* x z))) y)
       (if (<= z 4e+25) (fabs (/ (+ x 4.0) y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x * (z / y)));
	double tmp;
	if (z <= -1e+94) {
		tmp = t_0;
	} else if (z <= -48000.0) {
		tmp = (x + (4.0 - (x * z))) / y;
	} else if (z <= 4e+25) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    if (z <= (-1d+94)) then
        tmp = t_0
    else if (z <= (-48000.0d0)) then
        tmp = (x + (4.0d0 - (x * z))) / y
    else if (z <= 4d+25) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x * (z / y)));
	double tmp;
	if (z <= -1e+94) {
		tmp = t_0;
	} else if (z <= -48000.0) {
		tmp = (x + (4.0 - (x * z))) / y;
	} else if (z <= 4e+25) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x * (z / y)))
	tmp = 0
	if z <= -1e+94:
		tmp = t_0
	elif z <= -48000.0:
		tmp = (x + (4.0 - (x * z))) / y
	elif z <= 4e+25:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (z <= -1e+94)
		tmp = t_0;
	elseif (z <= -48000.0)
		tmp = Float64(Float64(x + Float64(4.0 - Float64(x * z))) / y);
	elseif (z <= 4e+25)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x * (z / y)));
	tmp = 0.0;
	if (z <= -1e+94)
		tmp = t_0;
	elseif (z <= -48000.0)
		tmp = (x + (4.0 - (x * z))) / y;
	elseif (z <= 4e+25)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1e+94], t$95$0, If[LessEqual[z, -48000.0], N[(N[(x + N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 4e+25], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e94 or 4.00000000000000036e25 < z

   1. Initial program 82.5%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in z around inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. associate-*r/ 78.9%

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

      3. *-commutative 78.9%

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

      4. distribute-rgt-neg-in 78.9%

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

   7. Simplified 78.9%

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

   if -1e94 < z < -48000

   1. Initial program 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. fmm-def 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. rem-square-sqrt 53.6%

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

      10. fabs-sqr 53.6%

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

      11. rem-square-sqrt 55.1%

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

      12. distribute-frac-neg2 55.1%

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

      13. distribute-frac-neg 55.1%

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

      14. neg-sub0 55.1%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 55.1%

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

   8. Taylor expanded in x around 0 55.1%

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

   if -48000 < z < 4.00000000000000036e25

   1. Initial program 93.6%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.4%

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

      1. +-commutative 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -48000:\\ \;\;\;\;\frac{x + \left(4 - x \cdot z\right)}{y}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 1.45 \cdot 10^{-34}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{-1}{y} + \frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.0) (not (<= x 1.45e-34)))
   (fabs (* x (/ (- 1.0 z) y)))
   (+ (* (* x z) (/ -1.0 y)) (/ (+ x 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.0) || !(x <= 1.45e-34)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = ((x * z) * (-1.0 / y)) + ((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)) .or. (.not. (x <= 1.45d-34))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = ((x * z) * ((-1.0d0) / y)) + ((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) || !(x <= 1.45e-34)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = ((x * z) * (-1.0 / y)) + ((x + 4.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.0) or not (x <= 1.45e-34):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = ((x * z) * (-1.0 / y)) + ((x + 4.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.0) || !(x <= 1.45e-34))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = Float64(Float64(Float64(x * z) * Float64(-1.0 / y)) + 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) || ~((x <= 1.45e-34)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = ((x * z) * (-1.0 / y)) + ((x + 4.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.0], N[Not[LessEqual[x, 1.45e-34]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * z), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4 or 1.4500000000000001e-34 < x

   1. Initial program 85.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around inf 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-/l* 98.9%

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

      3. associate-*r* 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*r/ 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

      8. distribute-lft-in 98.9%

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

      9. neg-mul-1 98.9%

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

      10. metadata-eval 98.9%

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

      11. +-commutative 98.9%

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

      12. unsub-neg 98.9%

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

   7. Simplified 98.9%

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

   if -4 < x < 1.4500000000000001e-34

   1. Initial program 94.6%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 50.4%

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

      2. fabs-sqr 50.4%

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

      3. add-sqr-sqrt 51.8%

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

      4. fma-undefine 51.8%

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

      5. distribute-rgt-in 51.8%

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

      6. sub-neg 51.8%

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

      7. metadata-eval 51.8%

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

      8. distribute-neg-in 51.8%

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

      9. +-commutative 51.8%

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

      10. frac-2neg 51.8%

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

      11. metadata-eval 51.8%

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

      12. div-inv 51.8%

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

      13. frac-2neg 51.8%

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

   6. Applied egg-rr 51.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 1.45 \cdot 10^{-34}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{-1}{y} + \frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fabs((((x * z) - (x + 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 = abs((((x * z) - (x + 4.0d0)) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. associate-*l/ 93.4%

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

   2. sub-div 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \left|\frac{x \cdot z - \left(x + 4\right)}{y}\right| \]
6. Add Preprocessing

Alternative 6: 48.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.088:\\ \;\;\;\;\frac{x \cdot z - x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - x \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.088)
   (/ (- (* x z) x) y)
   (if (<= x 4.0) (/ (- 4.0 (* x z)) y) (/ (- x (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.088) {
		tmp = ((x * z) - x) / y;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x - (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 <= (-0.088d0)) then
        tmp = ((x * z) - x) / y
    else if (x <= 4.0d0) then
        tmp = (4.0d0 - (x * z)) / y
    else
        tmp = (x - (x * z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.088) {
		tmp = ((x * z) - x) / y;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.088:
		tmp = ((x * z) - x) / y
	elif x <= 4.0:
		tmp = (4.0 - (x * z)) / y
	else:
		tmp = (x - (x * z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.088)
		tmp = Float64(Float64(Float64(x * z) - x) / y);
	elseif (x <= 4.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y);
	else
		tmp = Float64(Float64(x - Float64(x * z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.088)
		tmp = ((x * z) - x) / y;
	elseif (x <= 4.0)
		tmp = (4.0 - (x * z)) / y;
	else
		tmp = (x - (x * z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.088], N[(N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.087999999999999995

   1. Initial program 85.6%

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

      1. associate-*l/ 87.6%

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

      2. sub-div 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. add-sqr-sqrt 44.6%

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

      2. fabs-sqr 44.6%

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

      3. add-sqr-sqrt 45.1%

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

      4. frac-2neg 45.1%

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

      5. cancel-sign-sub-inv 45.1%

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

      6. *-commutative 45.1%

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

      7. distribute-neg-in 45.1%

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

      8. distribute-neg-in 45.1%

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

      9. add-sqr-sqrt 45.0%

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

      10. sqrt-unprod 35.6%

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

      11. sqr-neg 35.6%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

      15. distribute-rgt-neg-out 56.0%

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

      16. *-commutative 56.0%

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

      17. remove-double-neg 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 30.4%

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

      3. sqrt-unprod 57.7%

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

      4. sqr-neg 57.7%

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

      5. sqrt-unprod 30.2%

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

      6. add-sqr-sqrt 51.0%

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

      7. cancel-sign-sub-inv 51.0%

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

   8. Applied egg-rr 51.0%

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

   9. Taylor expanded in x around inf 51.0%

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

   if -0.087999999999999995 < x < 4

   1. Initial program 94.9%

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

      1. add-sqr-sqrt 49.4%

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

      2. fabs-sqr 49.4%

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

      3. add-sqr-sqrt 50.8%

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

      4. associate-*l/ 52.1%

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

      5. associate-/l* 47.3%

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in x around 0 46.9%

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

      1. associate-*r/ 51.8%

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

      2. sub-div 51.8%

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

      3. *-commutative 51.8%

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

   7. Applied egg-rr 51.8%

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

   if 4 < x

   1. Initial program 84.4%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 37.7%

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

      10. fabs-sqr 37.7%

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

      11. rem-square-sqrt 38.4%

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

      12. distribute-frac-neg2 38.4%

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

      13. distribute-frac-neg 38.4%

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

      14. neg-sub0 38.4%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 38.4%

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

   8. Taylor expanded in x around inf 37.2%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.088:\\ \;\;\;\;\frac{x \cdot z - x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -620:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - x \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -620.0)
   (/ (- 4.0 x) y)
   (if (<= x 4.0) (/ (- 4.0 (* x z)) y) (/ (- x (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -620.0) {
		tmp = (4.0 - x) / y;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x - (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 <= (-620.0d0)) then
        tmp = (4.0d0 - x) / y
    else if (x <= 4.0d0) then
        tmp = (4.0d0 - (x * z)) / y
    else
        tmp = (x - (x * z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -620.0) {
		tmp = (4.0 - x) / y;
	} else if (x <= 4.0) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -620.0:
		tmp = (4.0 - x) / y
	elif x <= 4.0:
		tmp = (4.0 - (x * z)) / y
	else:
		tmp = (x - (x * z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -620.0)
		tmp = Float64(Float64(4.0 - x) / y);
	elseif (x <= 4.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y);
	else
		tmp = Float64(Float64(x - Float64(x * z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -620.0)
		tmp = (4.0 - x) / y;
	elseif (x <= 4.0)
		tmp = (4.0 - (x * z)) / y;
	else
		tmp = (x - (x * z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -620.0], N[(N[(4.0 - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -620

   1. Initial program 85.6%

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

      1. associate-*l/ 87.6%

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

      2. sub-div 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. add-sqr-sqrt 44.6%

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

      2. fabs-sqr 44.6%

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

      3. add-sqr-sqrt 45.1%

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

      4. frac-2neg 45.1%

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

      5. cancel-sign-sub-inv 45.1%

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

      6. *-commutative 45.1%

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

      7. distribute-neg-in 45.1%

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

      8. distribute-neg-in 45.1%

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

      9. add-sqr-sqrt 45.0%

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

      10. sqrt-unprod 35.6%

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

      11. sqr-neg 35.6%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

      15. distribute-rgt-neg-out 56.0%

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

      16. *-commutative 56.0%

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

      17. remove-double-neg 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 30.4%

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

      3. sqrt-unprod 57.7%

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

      4. sqr-neg 57.7%

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

      5. sqrt-unprod 30.2%

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

      6. add-sqr-sqrt 51.0%

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

      7. cancel-sign-sub-inv 51.0%

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

   8. Applied egg-rr 51.0%

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

   9. Taylor expanded in z around 0 42.9%

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

       1. sub-neg 42.9%

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

       2. metadata-eval 42.9%

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

       3. neg-mul-1 42.9%

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

       4. distribute-frac-neg 42.9%

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

       5. neg-sub0 42.9%

          \[\leadsto \frac{\color{blue}{0 - \left(x + -4\right)}}{y} \]

       6. +-commutative 42.9%

          \[\leadsto \frac{0 - \color{blue}{\left(-4 + x\right)}}{y} \]

       7. associate--r+ 42.9%

          \[\leadsto \frac{\color{blue}{\left(0 - -4\right) - x}}{y} \]

       8. metadata-eval 42.9%

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

   11. Simplified 42.9%

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

   if -620 < x < 4

   1. Initial program 94.9%

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

      1. add-sqr-sqrt 49.4%

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

      2. fabs-sqr 49.4%

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

      3. add-sqr-sqrt 50.8%

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

      4. associate-*l/ 52.1%

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

      5. associate-/l* 47.3%

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in x around 0 46.9%

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

      1. associate-*r/ 51.8%

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

      2. sub-div 51.8%

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

      3. *-commutative 51.8%

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

   7. Applied egg-rr 51.8%

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

   if 4 < x

   1. Initial program 84.4%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 37.7%

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

      10. fabs-sqr 37.7%

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

      11. rem-square-sqrt 38.4%

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

      12. distribute-frac-neg2 38.4%

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

      13. distribute-frac-neg 38.4%

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

      14. neg-sub0 38.4%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 38.4%

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

   8. Taylor expanded in x around inf 37.2%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -620:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -520:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{elif}\;x \leq 88:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -520.0)
   (/ (- 4.0 x) y)
   (if (<= x 88.0) (/ (- 4.0 (* x z)) y) (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -520.0) {
		tmp = (4.0 - x) / y;
	} else if (x <= 88.0) {
		tmp = (4.0 - (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 <= (-520.0d0)) then
        tmp = (4.0d0 - x) / y
    else if (x <= 88.0d0) then
        tmp = (4.0d0 - (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 <= -520.0) {
		tmp = (4.0 - x) / y;
	} else if (x <= 88.0) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x - -4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -520.0:
		tmp = (4.0 - x) / y
	elif x <= 88.0:
		tmp = (4.0 - (x * z)) / y
	else:
		tmp = (x - -4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -520.0)
		tmp = Float64(Float64(4.0 - x) / y);
	elseif (x <= 88.0)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / 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 <= -520.0)
		tmp = (4.0 - x) / y;
	elseif (x <= 88.0)
		tmp = (4.0 - (x * z)) / y;
	else
		tmp = (x - -4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -520.0], N[(N[(4.0 - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 88.0], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -520

   1. Initial program 85.6%

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

      1. associate-*l/ 87.6%

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

      2. sub-div 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. add-sqr-sqrt 44.6%

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

      2. fabs-sqr 44.6%

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

      3. add-sqr-sqrt 45.1%

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

      4. frac-2neg 45.1%

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

      5. cancel-sign-sub-inv 45.1%

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

      6. *-commutative 45.1%

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

      7. distribute-neg-in 45.1%

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

      8. distribute-neg-in 45.1%

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

      9. add-sqr-sqrt 45.0%

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

      10. sqrt-unprod 35.6%

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

      11. sqr-neg 35.6%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

      15. distribute-rgt-neg-out 56.0%

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

      16. *-commutative 56.0%

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

      17. remove-double-neg 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 30.4%

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

      3. sqrt-unprod 57.7%

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

      4. sqr-neg 57.7%

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

      5. sqrt-unprod 30.2%

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

      6. add-sqr-sqrt 51.0%

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

      7. cancel-sign-sub-inv 51.0%

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

   8. Applied egg-rr 51.0%

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

   9. Taylor expanded in z around 0 42.9%

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

       1. sub-neg 42.9%

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

       2. metadata-eval 42.9%

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

       3. neg-mul-1 42.9%

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

       4. distribute-frac-neg 42.9%

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

       5. neg-sub0 42.9%

          \[\leadsto \frac{\color{blue}{0 - \left(x + -4\right)}}{y} \]

       6. +-commutative 42.9%

          \[\leadsto \frac{0 - \color{blue}{\left(-4 + x\right)}}{y} \]

       7. associate--r+ 42.9%

          \[\leadsto \frac{\color{blue}{\left(0 - -4\right) - x}}{y} \]

       8. metadata-eval 42.9%

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

   11. Simplified 42.9%

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

   if -520 < x < 88

   1. Initial program 94.9%

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

      1. add-sqr-sqrt 49.8%

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

      2. fabs-sqr 49.8%

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

      3. add-sqr-sqrt 51.1%

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

      4. associate-*l/ 52.5%

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

      5. associate-/l* 47.7%

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

   4. Applied egg-rr 47.7%

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

   5. Taylor expanded in x around 0 47.3%

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

      1. associate-*r/ 52.1%

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

      2. sub-div 52.2%

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

      3. *-commutative 52.2%

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

   7. Applied egg-rr 52.2%

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

   if 88 < x

   1. Initial program 84.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. distribute-neg-frac2 98.2%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 36.6%

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

      10. fabs-sqr 36.6%

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

      11. rem-square-sqrt 37.3%

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

      12. distribute-frac-neg2 37.3%

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

      13. distribute-frac-neg 37.3%

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

      14. neg-sub0 37.3%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 37.3%

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

   8. Taylor expanded in x around 0 33.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{elif}\;x \leq 88:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{elif}\;x \leq 215:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.4e-14)
   (/ (- 4.0 x) y)
   (if (<= x 215.0) (* z (/ x (- y))) (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.4e-14) {
		tmp = (4.0 - x) / y;
	} else if (x <= 215.0) {
		tmp = z * (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 <= 6.4d-14) then
        tmp = (4.0d0 - x) / y
    else if (x <= 215.0d0) then
        tmp = z * (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 <= 6.4e-14) {
		tmp = (4.0 - x) / y;
	} else if (x <= 215.0) {
		tmp = z * (x / -y);
	} else {
		tmp = (x - -4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.4e-14:
		tmp = (4.0 - x) / y
	elif x <= 215.0:
		tmp = z * (x / -y)
	else:
		tmp = (x - -4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.4e-14)
		tmp = Float64(Float64(4.0 - x) / y);
	elseif (x <= 215.0)
		tmp = Float64(z * Float64(x / 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 <= 6.4e-14)
		tmp = (4.0 - x) / y;
	elseif (x <= 215.0)
		tmp = z * (x / -y);
	else
		tmp = (x - -4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.4e-14], N[(N[(4.0 - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 215.0], N[(z * N[(x / (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.4000000000000005e-14

   1. Initial program 91.7%

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

      1. associate-*l/ 95.9%

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

      2. sub-div 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. add-sqr-sqrt 48.7%

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

      2. fabs-sqr 48.7%

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

      3. add-sqr-sqrt 49.9%

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

      4. frac-2neg 49.9%

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

      5. cancel-sign-sub-inv 49.9%

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

      6. *-commutative 49.9%

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

      7. distribute-neg-in 49.9%

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

      8. distribute-neg-in 49.9%

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

      9. add-sqr-sqrt 34.9%

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

      10. sqrt-unprod 46.8%

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

      11. sqr-neg 46.8%

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

      12. sqrt-unprod 14.9%

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

      13. add-sqr-sqrt 53.2%

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

      14. metadata-eval 53.2%

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

      15. distribute-rgt-neg-out 53.2%

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

      16. *-commutative 53.2%

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

      17. remove-double-neg 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. *-commutative 53.2%

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

      2. add-sqr-sqrt 25.4%

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

      3. sqrt-unprod 48.4%

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

      4. sqr-neg 48.4%

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

      5. sqrt-unprod 29.3%

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

      6. add-sqr-sqrt 48.1%

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

      7. cancel-sign-sub-inv 48.1%

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

   8. Applied egg-rr 48.1%

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

   9. Taylor expanded in z around 0 40.5%

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

       1. sub-neg 40.5%

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

       2. metadata-eval 40.5%

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

       3. neg-mul-1 40.5%

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

       4. distribute-frac-neg 40.5%

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

       5. neg-sub0 40.5%

          \[\leadsto \frac{\color{blue}{0 - \left(x + -4\right)}}{y} \]

       6. +-commutative 40.5%

          \[\leadsto \frac{0 - \color{blue}{\left(-4 + x\right)}}{y} \]

       7. associate--r+ 40.5%

          \[\leadsto \frac{\color{blue}{\left(0 - -4\right) - x}}{y} \]

       8. metadata-eval 40.5%

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

   11. Simplified 40.5%

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

   if 6.4000000000000005e-14 < x < 215

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

      4. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. add-sqr-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. add-sqr-sqrt 57.2%

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

      4. associate-/l* 57.2%

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

      5. distribute-frac-neg 57.2%

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

      6. add-sqr-sqrt 56.9%

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

      7. sqrt-unprod 57.2%

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

      8. sqr-neg 57.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 42.5%

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

      11. distribute-rgt-neg-in 42.5%

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

      12. distribute-lft-neg-in 42.5%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 57.2%

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

      15. sqr-neg 57.2%

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

      16. sqrt-unprod 56.9%

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

      17. add-sqr-sqrt 57.2%

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

   9. Applied egg-rr 57.2%

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

   if 215 < x

   1. Initial program 84.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. distribute-neg-frac2 98.2%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 36.6%

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

      10. fabs-sqr 36.6%

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

      11. rem-square-sqrt 37.3%

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

      12. distribute-frac-neg2 37.3%

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

      13. distribute-frac-neg 37.3%

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

      14. neg-sub0 37.3%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 37.3%

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

   8. Taylor expanded in x around 0 33.6%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{elif}\;x \leq 215:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1:\\ \;\;\;\;\frac{x \cdot z - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \left(x \cdot z - 4\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1) (/ (- (* x z) x) y) (/ (- x (- (* x z) 4.0)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1) {
		tmp = ((x * z) - x) / y;
	} else {
		tmp = (x - ((x * z) - 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 <= (-3.1d0)) then
        tmp = ((x * z) - x) / y
    else
        tmp = (x - ((x * z) - 4.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1) {
		tmp = ((x * z) - x) / y;
	} else {
		tmp = (x - ((x * z) - 4.0)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1:
		tmp = ((x * z) - x) / y
	else:
		tmp = (x - ((x * z) - 4.0)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1)
		tmp = Float64(Float64(Float64(x * z) - x) / y);
	else
		tmp = Float64(Float64(x - Float64(Float64(x * z) - 4.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1)
		tmp = ((x * z) - x) / y;
	else
		tmp = (x - ((x * z) - 4.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1], N[(N[(N[(x * z), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision], N[(N[(x - N[(N[(x * z), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000009

   1. Initial program 85.6%

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

      1. associate-*l/ 87.6%

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

      2. sub-div 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. add-sqr-sqrt 44.6%

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

      2. fabs-sqr 44.6%

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

      3. add-sqr-sqrt 45.1%

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

      4. frac-2neg 45.1%

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

      5. cancel-sign-sub-inv 45.1%

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

      6. *-commutative 45.1%

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

      7. distribute-neg-in 45.1%

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

      8. distribute-neg-in 45.1%

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

      9. add-sqr-sqrt 45.0%

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

      10. sqrt-unprod 35.6%

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

      11. sqr-neg 35.6%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

      15. distribute-rgt-neg-out 56.0%

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

      16. *-commutative 56.0%

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

      17. remove-double-neg 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 30.4%

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

      3. sqrt-unprod 57.7%

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

      4. sqr-neg 57.7%

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

      5. sqrt-unprod 30.2%

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

      6. add-sqr-sqrt 51.0%

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

      7. cancel-sign-sub-inv 51.0%

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

   8. Applied egg-rr 51.0%

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

   9. Taylor expanded in x around inf 51.0%

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

   if -3.10000000000000009 < x

   1. Initial program 91.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-neg-frac2 99.4%

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

      4. fmm-def 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. sub-neg 99.4%

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

      9. rem-square-sqrt 46.8%

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

      10. fabs-sqr 46.8%

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

      11. rem-square-sqrt 48.0%

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

      12. distribute-frac-neg2 48.0%

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

      13. distribute-frac-neg 48.0%

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

      14. neg-sub0 48.0%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 48.0%

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

   8. Taylor expanded in x around 0 48.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1:\\ \;\;\;\;\frac{x \cdot z - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \left(x \cdot z - 4\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.5e-5) (* z (/ x y)) (if (<= x 4.0) (/ 4.0 y) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e-5) {
		tmp = z * (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 <= (-8.5d-5)) then
        tmp = z * (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 <= -8.5e-5) {
		tmp = z * (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 <= -8.5e-5:
		tmp = z * (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 <= -8.5e-5)
		tmp = Float64(z * 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 <= -8.5e-5)
		tmp = z * (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, -8.5e-5], N[(z * N[(x / y), $MachinePrecision]), $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 < -8.500000000000001e-5

   1. Initial program 85.6%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in z around inf 54.9%

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

      1. mul-1-neg 54.9%

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

      2. distribute-frac-neg 54.9%

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

      3. distribute-lft-neg-in 54.9%

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

      4. *-commutative 54.9%

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

   7. Simplified 54.9%

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

      1. add-sqr-sqrt 30.2%

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

      2. fabs-sqr 30.2%

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

      3. add-sqr-sqrt 30.6%

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

      4. associate-/l* 39.3%

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

      5. *-commutative 39.3%

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

      6. add-sqr-sqrt 39.3%

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

      7. sqrt-unprod 33.4%

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

      8. sqr-neg 33.4%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 32.7%

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

   9. Applied egg-rr 32.7%

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

   if -8.500000000000001e-5 < x < 4

   1. Initial program 94.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

      9. rem-square-sqrt 50.7%

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

      10. fabs-sqr 50.7%

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

      11. rem-square-sqrt 52.1%

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

      12. distribute-frac-neg2 52.1%

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

      13. distribute-frac-neg 52.1%

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

      14. neg-sub0 52.1%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 52.1%

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

   8. Taylor expanded in x around 0 37.8%

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

   if 4 < x

   1. Initial program 84.4%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 37.7%

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

      10. fabs-sqr 37.7%

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

      11. rem-square-sqrt 38.4%

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

      12. distribute-frac-neg2 38.4%

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

      13. distribute-frac-neg 38.4%

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

      14. neg-sub0 38.4%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 38.4%

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

   8. Taylor expanded in x around 0 33.1%

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

   9. Taylor expanded in x around inf 31.9%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.1% 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 85.6%

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

      1. associate-*l/ 87.6%

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

      2. sub-div 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. add-sqr-sqrt 44.6%

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

      2. fabs-sqr 44.6%

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

      3. add-sqr-sqrt 45.1%

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

      4. frac-2neg 45.1%

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

      5. cancel-sign-sub-inv 45.1%

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

      6. *-commutative 45.1%

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

      7. distribute-neg-in 45.1%

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

      8. distribute-neg-in 45.1%

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

      9. add-sqr-sqrt 45.0%

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

      10. sqrt-unprod 35.6%

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

      11. sqr-neg 35.6%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

      15. distribute-rgt-neg-out 56.0%

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

      16. *-commutative 56.0%

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

      17. remove-double-neg 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 30.4%

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

      3. sqrt-unprod 57.7%

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

      4. sqr-neg 57.7%

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

      5. sqrt-unprod 30.2%

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

      6. add-sqr-sqrt 51.0%

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

      7. cancel-sign-sub-inv 51.0%

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

   8. Applied egg-rr 51.0%

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

   9. Taylor expanded in z around 0 42.9%

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

       1. sub-neg 42.9%

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

       2. metadata-eval 42.9%

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

       3. neg-mul-1 42.9%

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

       4. distribute-frac-neg 42.9%

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

       5. neg-sub0 42.9%

          \[\leadsto \frac{\color{blue}{0 - \left(x + -4\right)}}{y} \]

       6. +-commutative 42.9%

          \[\leadsto \frac{0 - \color{blue}{\left(-4 + x\right)}}{y} \]

       7. associate--r+ 42.9%

          \[\leadsto \frac{\color{blue}{\left(0 - -4\right) - x}}{y} \]

       8. metadata-eval 42.9%

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

   11. Simplified 42.9%

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

   if -4 < x

   1. Initial program 91.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-neg-frac2 99.4%

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

      4. fmm-def 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. sub-neg 99.4%

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

      9. rem-square-sqrt 46.8%

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

      10. fabs-sqr 46.8%

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

      11. rem-square-sqrt 48.0%

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

      12. distribute-frac-neg2 48.0%

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

      13. distribute-frac-neg 48.0%

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

      14. neg-sub0 48.0%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 48.0%

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

   8. Taylor expanded in x around 0 36.6%

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

Alternative 13: 34.8% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 2.0) (/ (- 4.0 x) y) (/ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.0) {
		tmp = (4.0 - x) / 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 <= 2.0d0) then
        tmp = (4.0d0 - x) / 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 <= 2.0) {
		tmp = (4.0 - x) / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.0:
		tmp = (4.0 - x) / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(4.0 - x) / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (4.0 - x) / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.0], N[(N[(4.0 - x), $MachinePrecision] / y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 91.9%

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

      1. associate-*l/ 96.0%

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

      2. sub-div 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. add-sqr-sqrt 48.8%

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

      2. fabs-sqr 48.8%

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

      3. add-sqr-sqrt 49.9%

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

      4. frac-2neg 49.9%

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

      5. cancel-sign-sub-inv 49.9%

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

      6. *-commutative 49.9%

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

      7. distribute-neg-in 49.9%

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

      8. distribute-neg-in 49.9%

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

      9. add-sqr-sqrt 33.9%

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

      10. sqrt-unprod 46.9%

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

      11. sqr-neg 46.9%

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

      12. sqrt-unprod 16.0%

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

      13. add-sqr-sqrt 53.1%

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

      14. metadata-eval 53.1%

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

      15. distribute-rgt-neg-out 53.1%

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

      16. *-commutative 53.1%

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

      17. remove-double-neg 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. *-commutative 53.1%

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

      2. add-sqr-sqrt 25.1%

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

      3. sqrt-unprod 47.9%

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

      4. sqr-neg 47.9%

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

      5. sqrt-unprod 29.4%

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

      6. add-sqr-sqrt 48.1%

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

      7. cancel-sign-sub-inv 48.1%

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

   8. Applied egg-rr 48.1%

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

   9. Taylor expanded in z around 0 39.4%

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

       1. sub-neg 39.4%

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

       2. metadata-eval 39.4%

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

       3. neg-mul-1 39.4%

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

       4. distribute-frac-neg 39.4%

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

       5. neg-sub0 39.4%

          \[\leadsto \frac{\color{blue}{0 - \left(x + -4\right)}}{y} \]

       6. +-commutative 39.4%

          \[\leadsto \frac{0 - \color{blue}{\left(-4 + x\right)}}{y} \]

       7. associate--r+ 39.4%

          \[\leadsto \frac{\color{blue}{\left(0 - -4\right) - x}}{y} \]

       8. metadata-eval 39.4%

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

   11. Simplified 39.4%

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

   if 2 < x

   1. Initial program 84.4%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 37.7%

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

      10. fabs-sqr 37.7%

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

      11. rem-square-sqrt 38.4%

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

      12. distribute-frac-neg2 38.4%

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

      13. distribute-frac-neg 38.4%

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

      14. neg-sub0 38.4%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 38.4%

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

   8. Taylor expanded in x around 0 33.1%

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

   9. Taylor expanded in x around inf 31.9%

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

Alternative 14: 27.9% 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 91.9%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. fmm-def 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. +-commutative 98.5%

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

      8. sub-neg 98.5%

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

      9. rem-square-sqrt 48.8%

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

      10. fabs-sqr 48.8%

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

      11. rem-square-sqrt 49.9%

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

      12. distribute-frac-neg2 49.9%

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

      13. distribute-frac-neg 49.9%

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

      14. neg-sub0 49.9%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 49.9%

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

   8. Taylor expanded in x around 0 26.9%

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

   if 4 < x

   1. Initial program 84.4%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. fmm-def 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. sub-neg 98.2%

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

      9. rem-square-sqrt 37.7%

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

      10. fabs-sqr 37.7%

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

      11. rem-square-sqrt 38.4%

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

      12. distribute-frac-neg2 38.4%

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

      13. distribute-frac-neg 38.4%

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

      14. neg-sub0 38.4%

          \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

   7. Simplified 38.4%

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

   8. Taylor expanded in x around 0 33.1%

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

   9. Taylor expanded in x around inf 31.9%

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

Alternative 15: 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%

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

3. Simplified 98.4%

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

5. Taylor expanded in y around 0 98.4%

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

   1. mul-1-neg 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. fmm-def 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. sub-neg 98.4%

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

   9. rem-square-sqrt 46.3%

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

   10. fabs-sqr 46.3%

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

   11. rem-square-sqrt 47.3%

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

   12. distribute-frac-neg2 47.3%

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

   13. distribute-frac-neg 47.3%

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

   14. neg-sub0 47.3%

       \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(x, z, -4 - x\right)}}{y} \]

7. Simplified 47.3%

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

8. Taylor expanded in x around 0 21.6%

   \[\leadsto \frac{\color{blue}{4}}{y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024188 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))