fabs fraction 1

Percentage Accurate: 92.0% → 99.0%

Time: 10.7s

Alternatives: 18

Speedup: 0.5×

• Rival profile vector overflowed, profile may not be complete

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \left|t\_0 - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\left|t\_0 - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-1 + z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)) (t_1 (fabs (- t_0 (* z (/ x y))))))
   (if (<= t_1 2e-124)
     (fabs (- t_0 (/ x (/ y z))))
     (if (<= t_1 2e+299) t_1 (fabs (* (/ x y) (+ -1.0 z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = fabs((t_0 - (z * (x / y))));
	double tmp;
	if (t_1 <= 2e-124) {
		tmp = fabs((t_0 - (x / (y / z))));
	} else if (t_1 <= 2e+299) {
		tmp = t_1;
	} else {
		tmp = fabs(((x / y) * (-1.0 + z)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y
    t_1 = abs((t_0 - (z * (x / y))))
    if (t_1 <= 2d-124) then
        tmp = abs((t_0 - (x / (y / z))))
    else if (t_1 <= 2d+299) then
        tmp = t_1
    else
        tmp = abs(((x / y) * ((-1.0d0) + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = Math.abs((t_0 - (z * (x / y))));
	double tmp;
	if (t_1 <= 2e-124) {
		tmp = Math.abs((t_0 - (x / (y / z))));
	} else if (t_1 <= 2e+299) {
		tmp = t_1;
	} else {
		tmp = Math.abs(((x / y) * (-1.0 + z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + 4.0) / y
	t_1 = math.fabs((t_0 - (z * (x / y))))
	tmp = 0
	if t_1 <= 2e-124:
		tmp = math.fabs((t_0 - (x / (y / z))))
	elif t_1 <= 2e+299:
		tmp = t_1
	else:
		tmp = math.fabs(((x / y) * (-1.0 + z)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	t_1 = abs(Float64(t_0 - Float64(z * Float64(x / y))))
	tmp = 0.0
	if (t_1 <= 2e-124)
		tmp = abs(Float64(t_0 - Float64(x / Float64(y / z))));
	elseif (t_1 <= 2e+299)
		tmp = t_1;
	else
		tmp = abs(Float64(Float64(x / y) * Float64(-1.0 + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + 4.0) / y;
	t_1 = abs((t_0 - (z * (x / y))));
	tmp = 0.0;
	if (t_1 <= 2e-124)
		tmp = abs((t_0 - (x / (y / z))));
	elseif (t_1 <= 2e+299)
		tmp = t_1;
	else
		tmp = abs(((x / y) * (-1.0 + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(t$95$0 - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-124], N[Abs[N[(t$95$0 - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+299], t$95$1, N[Abs[N[(N[(x / y), $MachinePrecision] * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.99999999999999987e-124

   1. Initial program 83.7%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.99999999999999987e-124 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2.0000000000000001e299

   1. Initial program 99.9%

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

   if 2.0000000000000001e299 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 71.2%

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

      1. fabs-sub 71.2%

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

      2. associate-*l/ 75.0%

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

      3. associate-*r/ 75.0%

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

      4. fmm-def 80.8%

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

      5. distribute-neg-frac 80.8%

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

      6. +-commutative 80.8%

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

      7. distribute-neg-in 80.8%

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

      8. unsub-neg 80.8%

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

      9. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. div-sub 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. neg-mul-1 100.0%

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

      15. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-1 + z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-1 + z\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* z (/ x y))))))
   (if (<= t_0 2e+299) t_0 (fabs (* (/ x y) (+ -1.0 z))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((x + 4.0) / y) - (z * (x / y))));
	double tmp;
	if (t_0 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = fabs(((x / y) * (-1.0 + z)));
	}
	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 + 4.0d0) / y) - (z * (x / y))))
    if (t_0 <= 2d+299) then
        tmp = t_0
    else
        tmp = abs(((x / y) * ((-1.0d0) + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((((x + 4.0) / y) - (z * (x / y))));
	double tmp;
	if (t_0 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((x / y) * (-1.0 + z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((((x + 4.0) / y) - (z * (x / y))))
	tmp = 0
	if t_0 <= 2e+299:
		tmp = t_0
	else:
		tmp = math.fabs(((x / y) * (-1.0 + z)))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y))))
	tmp = 0.0
	if (t_0 <= 2e+299)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(x / y) * Float64(-1.0 + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((((x + 4.0) / y) - (z * (x / y))));
	tmp = 0.0;
	if (t_0 <= 2e+299)
		tmp = t_0;
	else
		tmp = abs(((x / y) * (-1.0 + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e+299], t$95$0, N[Abs[N[(N[(x / y), $MachinePrecision] * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2.0000000000000001e299

   1. Initial program 97.0%

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

   if 2.0000000000000001e299 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 71.2%

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

      1. fabs-sub 71.2%

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

      2. associate-*l/ 75.0%

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

      3. associate-*r/ 75.0%

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

      4. fmm-def 80.8%

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

      5. distribute-neg-frac 80.8%

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

      6. +-commutative 80.8%

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

      7. distribute-neg-in 80.8%

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

      8. unsub-neg 80.8%

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

      9. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. div-sub 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. neg-mul-1 100.0%

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

      15. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-1 + z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-58}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.4e-58)
   (fabs (* (/ -1.0 y) (fma x z (- -4.0 x))))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.4e-58) {
		tmp = fabs(((-1.0 / y) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.4e-58)
		tmp = abs(Float64(Float64(-1.0 / y) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.4e-58], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.39999999999999973e-58

   1. Initial program 89.9%

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

   3. Simplified 97.8%

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

   if 3.39999999999999973e-58 < y

   1. Initial program 96.2%

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

      1. fabs-sub 96.2%

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

      2. associate-*l/ 91.1%

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

      3. associate-*r/ 99.8%

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

      4. fmm-def 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

Alternative 4: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-76}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e-76)
   (fabs (* (/ -1.0 y) (fma x z (- -4.0 x))))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e-76) {
		tmp = fabs(((-1.0 / y) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e-76)
		tmp = abs(Float64(Float64(-1.0 / y) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e-76], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e-76

   1. Initial program 89.5%

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

   3. Simplified 97.7%

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

   if 1.3e-76 < y

   1. Initial program 96.5%

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

      1. associate-*l/ 91.8%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

Alternative 5: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-11} \lor \neg \left(x \leq 3 \cdot 10^{-11}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-1 + z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{y} \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.3e-11) (not (<= x 3e-11)))
   (fabs (* (/ x y) (+ -1.0 z)))
   (+ (/ (+ x 4.0) y) (* (/ -1.0 y) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.3e-11) || !(x <= 3e-11)) {
		tmp = fabs(((x / y) * (-1.0 + z)));
	} else {
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z));
	}
	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 <= (-3.3d-11)) .or. (.not. (x <= 3d-11))) then
        tmp = abs(((x / y) * ((-1.0d0) + z)))
    else
        tmp = ((x + 4.0d0) / y) + (((-1.0d0) / y) * (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.3e-11) || !(x <= 3e-11)) {
		tmp = Math.abs(((x / y) * (-1.0 + z)));
	} else {
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.3e-11) or not (x <= 3e-11):
		tmp = math.fabs(((x / y) * (-1.0 + z)))
	else:
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.3e-11) || !(x <= 3e-11))
		tmp = abs(Float64(Float64(x / y) * Float64(-1.0 + z)));
	else
		tmp = Float64(Float64(Float64(x + 4.0) / y) + Float64(Float64(-1.0 / y) * Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.3e-11) || ~((x <= 3e-11)))
		tmp = abs(((x / y) * (-1.0 + z)));
	else
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.3e-11], N[Not[LessEqual[x, 3e-11]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] + N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3000000000000002e-11 or 3e-11 < x

   1. Initial program 88.9%

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

      1. fabs-sub 88.9%

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

      2. associate-*l/ 82.8%

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

      3. associate-*r/ 90.4%

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

      4. fmm-def 92.6%

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

      5. distribute-neg-frac 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. unsub-neg 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. div-sub 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 91.7%

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

      4. associate-/l* 99.3%

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

      5. sub-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. neg-mul-1 99.3%

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

      8. metadata-eval 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-in 99.3%

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

      11. +-commutative 99.3%

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

      12. distribute-lft-in 99.3%

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

      13. metadata-eval 99.3%

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

      14. neg-mul-1 99.3%

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

      15. remove-double-neg 99.3%

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

   7. Simplified 99.3%

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

   if -3.3000000000000002e-11 < x < 3e-11

   1. Initial program 95.0%

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

   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 45.9%

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

      2. fabs-sqr 45.9%

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

      3. add-sqr-sqrt 47.2%

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

      4. fma-undefine 47.2%

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

      5. distribute-rgt-in 47.2%

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

      6. sub-neg 47.2%

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

      7. metadata-eval 47.2%

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

      8. distribute-neg-in 47.2%

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

      9. +-commutative 47.2%

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

      10. frac-2neg 47.2%

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

      11. metadata-eval 47.2%

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

      12. div-inv 47.2%

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

      13. frac-2neg 47.2%

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

   6. Applied egg-rr 47.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-11} \lor \neg \left(x \leq 3 \cdot 10^{-11}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(-1 + z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{y} \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-1 + z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 10^{+31}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{y} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0)
   (/ (+ -1.0 z) (/ y x))
   (if (<= x 1e+31)
     (+ (/ (+ x 4.0) y) (* (/ -1.0 y) (* x z)))
     (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-1.0 + z) / (y / x);
	} else if (x <= 1e+31) {
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z));
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	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 = ((-1.0d0) + z) / (y / x)
    else if (x <= 1d+31) then
        tmp = ((x + 4.0d0) / y) + (((-1.0d0) / y) * (x * z))
    else
        tmp = (x / y) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-1.0 + z) / (y / x);
	} else if (x <= 1e+31) {
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z));
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = (-1.0 + z) / (y / x)
	elif x <= 1e+31:
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z))
	else:
		tmp = (x / y) * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-1.0 + z) / Float64(y / x));
	elseif (x <= 1e+31)
		tmp = Float64(Float64(Float64(x + 4.0) / y) + Float64(Float64(-1.0 / y) * Float64(x * z)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.0)
		tmp = (-1.0 + z) / (y / x);
	elseif (x <= 1e+31)
		tmp = ((x + 4.0) / y) + ((-1.0 / y) * (x * z));
	else
		tmp = (x / y) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.0], N[(N[(-1.0 + z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+31], N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] + N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4

   1. Initial program 87.4%

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

      1. fabs-sub 87.4%

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

      2. associate-*l/ 80.6%

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

      3. associate-*r/ 87.4%

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

      4. fmm-def 90.9%

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

      5. distribute-neg-frac 90.9%

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

      6. +-commutative 90.9%

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

      7. distribute-neg-in 90.9%

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

      8. unsub-neg 90.9%

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

      9. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. div-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 93.1%

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

      4. associate-/l* 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. neg-mul-1 100.0%

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

      15. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. fabs-sqr 64.0%

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

      3. add-sqr-sqrt 64.5%

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

      4. +-commutative 64.5%

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

      5. clear-num 64.4%

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

      6. un-div-inv 64.4%

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

      7. +-commutative 64.4%

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

   9. Applied egg-rr 64.4%

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

   if -4 < x < 9.9999999999999996e30

   1. Initial program 95.5%

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

   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. Step-by-step derivation

      1. add-sqr-sqrt 45.6%

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

      2. fabs-sqr 45.6%

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

      3. add-sqr-sqrt 46.9%

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

      4. fma-undefine 46.9%

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

      5. distribute-rgt-in 46.9%

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

      6. sub-neg 46.9%

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

      7. metadata-eval 46.9%

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

      8. distribute-neg-in 46.9%

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

      9. +-commutative 46.9%

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

      10. frac-2neg 46.9%

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

      11. metadata-eval 46.9%

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

      12. div-inv 46.9%

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

      13. frac-2neg 46.9%

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

   6. Applied egg-rr 46.9%

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

   if 9.9999999999999996e30 < x

   1. Initial program 87.6%

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

      1. fabs-sub 87.6%

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

      2. associate-*l/ 80.6%

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

      3. associate-*r/ 90.7%

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

      4. fmm-def 92.2%

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

      5. distribute-neg-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. distribute-neg-in 92.2%

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

      8. unsub-neg 92.2%

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

      9. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

      1. fma-undefine 90.7%

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

      2. associate-*r/ 80.6%

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

      3. associate-*l/ 87.6%

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

      4. div-inv 87.5%

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

      5. sub-neg 87.5%

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

      6. metadata-eval 87.5%

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

      7. distribute-neg-in 87.5%

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

      8. +-commutative 87.5%

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

      9. cancel-sign-sub-inv 87.5%

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

      10. div-inv 87.6%

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

      11. fabs-sub 87.6%

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

      12. add-sqr-sqrt 48.8%

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

      13. fabs-sqr 48.8%

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

      14. add-sqr-sqrt 49.4%

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

      15. sub-neg 49.4%

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

      16. distribute-rgt-neg-in 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in x around inf 49.4%

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

      1. distribute-rgt-neg-out 49.4%

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

      2. distribute-lft-neg-in 49.4%

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

      3. add-sqr-sqrt 14.0%

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

      4. sqrt-unprod 26.9%

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

      5. sqr-neg 26.9%

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

      6. sqrt-unprod 18.7%

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

      7. add-sqr-sqrt 41.8%

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

      8. cancel-sign-sub-inv 41.8%

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

      9. *-un-lft-identity 41.8%

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

      10. *-commutative 41.8%

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

      11. distribute-rgt-out-- 51.1%

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

      12. add-sqr-sqrt 27.9%

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

      13. sqrt-unprod 42.3%

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

      14. sqr-neg 42.3%

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

      15. sqrt-unprod 20.1%

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

      16. add-sqr-sqrt 55.6%

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

   9. Applied egg-rr 55.6%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-1 + z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 10^{+31}:\\ \;\;\;\;\frac{x + 4}{y} + \frac{-1}{y} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8:\\ \;\;\;\;\frac{-1 + z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8)
   (/ (+ -1.0 z) (/ y x))
   (if (<= x 5e+25) (/ (- (+ x 4.0) (* x z)) y) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8) {
		tmp = (-1.0 + z) / (y / x);
	} else if (x <= 5e+25) {
		tmp = ((x + 4.0) - (x * z)) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	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 <= (-5.8d0)) then
        tmp = ((-1.0d0) + z) / (y / x)
    else if (x <= 5d+25) then
        tmp = ((x + 4.0d0) - (x * z)) / y
    else
        tmp = (x / y) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8) {
		tmp = (-1.0 + z) / (y / x);
	} else if (x <= 5e+25) {
		tmp = ((x + 4.0) - (x * z)) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8:
		tmp = (-1.0 + z) / (y / x)
	elif x <= 5e+25:
		tmp = ((x + 4.0) - (x * z)) / y
	else:
		tmp = (x / y) * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.8)
		tmp = Float64(Float64(-1.0 + z) / Float64(y / x));
	elseif (x <= 5e+25)
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.8)
		tmp = (-1.0 + z) / (y / x);
	elseif (x <= 5e+25)
		tmp = ((x + 4.0) - (x * z)) / y;
	else
		tmp = (x / y) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8], N[(N[(-1.0 + z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+25], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.79999999999999982

   1. Initial program 87.4%

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

      1. fabs-sub 87.4%

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

      2. associate-*l/ 80.6%

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

      3. associate-*r/ 87.4%

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

      4. fmm-def 90.9%

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

      5. distribute-neg-frac 90.9%

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

      6. +-commutative 90.9%

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

      7. distribute-neg-in 90.9%

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

      8. unsub-neg 90.9%

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

      9. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. div-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 93.1%

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

      4. associate-/l* 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. neg-mul-1 100.0%

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

      15. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. fabs-sqr 64.0%

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

      3. add-sqr-sqrt 64.5%

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

      4. +-commutative 64.5%

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

      5. clear-num 64.4%

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

      6. un-div-inv 64.4%

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

      7. +-commutative 64.4%

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

   9. Applied egg-rr 64.4%

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

   if -5.79999999999999982 < x < 5.00000000000000024e25

   1. Initial program 95.4%

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

      1. fabs-sub 95.4%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 92.9%

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

      5. distribute-neg-frac 92.9%

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

      6. +-commutative 92.9%

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

      7. distribute-neg-in 92.9%

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

      8. unsub-neg 92.9%

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

      9. metadata-eval 92.9%

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

   3. Simplified 92.9%

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

      1. fma-undefine 92.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.4%

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

      4. div-inv 95.4%

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

      5. sub-neg 95.4%

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

      6. metadata-eval 95.4%

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

      7. distribute-neg-in 95.4%

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

      8. +-commutative 95.4%

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

      9. cancel-sign-sub-inv 95.4%

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

      10. div-inv 95.4%

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

      11. fabs-sub 95.4%

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

      12. add-sqr-sqrt 42.0%

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

      13. fabs-sqr 42.0%

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

      14. add-sqr-sqrt 43.3%

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

      15. associate-*l/ 46.8%

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

      16. sub-div 46.8%

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

   6. Applied egg-rr 46.8%

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

   if 5.00000000000000024e25 < x

   1. Initial program 88.0%

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

      1. fabs-sub 88.0%

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

      2. associate-*l/ 81.2%

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

      3. associate-*r/ 91.0%

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

      4. fmm-def 92.4%

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

      5. distribute-neg-frac 92.4%

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

      6. +-commutative 92.4%

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

      7. distribute-neg-in 92.4%

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

      8. unsub-neg 92.4%

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

      9. metadata-eval 92.4%

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

   3. Simplified 92.4%

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

      1. fma-undefine 91.0%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 88.0%

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

      4. div-inv 87.8%

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

      5. sub-neg 87.8%

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

      6. metadata-eval 87.8%

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

      7. distribute-neg-in 87.8%

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

      8. +-commutative 87.8%

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

      9. cancel-sign-sub-inv 87.8%

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

      10. div-inv 88.0%

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

      11. fabs-sub 88.0%

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

      12. add-sqr-sqrt 48.8%

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

      13. fabs-sqr 48.8%

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

      14. add-sqr-sqrt 49.5%

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

      15. sub-neg 49.5%

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

      16. distribute-rgt-neg-in 49.5%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in x around inf 49.5%

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

      1. distribute-rgt-neg-out 49.5%

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

      2. distribute-lft-neg-in 49.5%

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

      3. add-sqr-sqrt 15.0%

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

      4. sqrt-unprod 27.6%

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

      5. sqr-neg 27.6%

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

      6. sqrt-unprod 18.1%

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

      7. add-sqr-sqrt 40.6%

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

      8. cancel-sign-sub-inv 40.6%

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

      9. *-un-lft-identity 40.6%

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

      10. *-commutative 40.6%

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

      11. distribute-rgt-out-- 49.6%

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

      12. add-sqr-sqrt 27.1%

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

      13. sqrt-unprod 42.5%

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

      14. sqr-neg 42.5%

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

      15. sqrt-unprod 21.0%

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

      16. add-sqr-sqrt 55.5%

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

   9. Applied egg-rr 55.5%

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

Alternative 8: 50.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1350:\\ \;\;\;\;\frac{-1 + z}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 0.000225:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1350.0)
   (/ (+ -1.0 z) (/ y x))
   (if (<= x 0.000225) (/ (- 4.0 (* x z)) y) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1350.0) {
		tmp = (-1.0 + z) / (y / x);
	} else if (x <= 0.000225) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	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 <= (-1350.0d0)) then
        tmp = ((-1.0d0) + z) / (y / x)
    else if (x <= 0.000225d0) then
        tmp = (4.0d0 - (x * z)) / y
    else
        tmp = (x / y) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1350.0) {
		tmp = (-1.0 + z) / (y / x);
	} else if (x <= 0.000225) {
		tmp = (4.0 - (x * z)) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1350.0:
		tmp = (-1.0 + z) / (y / x)
	elif x <= 0.000225:
		tmp = (4.0 - (x * z)) / y
	else:
		tmp = (x / y) * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1350.0)
		tmp = Float64(Float64(-1.0 + z) / Float64(y / x));
	elseif (x <= 0.000225)
		tmp = Float64(Float64(4.0 - Float64(x * z)) / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1350.0)
		tmp = (-1.0 + z) / (y / x);
	elseif (x <= 0.000225)
		tmp = (4.0 - (x * z)) / y;
	else
		tmp = (x / y) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1350.0], N[(N[(-1.0 + z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000225], N[(N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1350

   1. Initial program 87.4%

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

      1. fabs-sub 87.4%

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

      2. associate-*l/ 80.6%

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

      3. associate-*r/ 87.4%

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

      4. fmm-def 90.9%

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

      5. distribute-neg-frac 90.9%

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

      6. +-commutative 90.9%

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

      7. distribute-neg-in 90.9%

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

      8. unsub-neg 90.9%

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

      9. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. div-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 93.1%

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

      4. associate-/l* 100.0%

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

      5. sub-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. neg-mul-1 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. neg-mul-1 100.0%

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

      15. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. fabs-sqr 64.0%

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

      3. add-sqr-sqrt 64.5%

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

      4. +-commutative 64.5%

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

      5. clear-num 64.4%

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

      6. un-div-inv 64.4%

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

      7. +-commutative 64.4%

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

   9. Applied egg-rr 64.4%

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

   if -1350 < x < 2.2499999999999999e-4

   1. Initial program 95.1%

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

      1. fabs-sub 95.1%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 92.4%

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

      4. fmm-def 92.4%

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

      5. distribute-neg-frac 92.4%

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

      6. +-commutative 92.4%

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

      7. distribute-neg-in 92.4%

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

      8. unsub-neg 92.4%

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

      9. metadata-eval 92.4%

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

   3. Simplified 92.4%

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

      1. fma-undefine 92.4%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.1%

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

      4. div-inv 95.1%

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

      5. sub-neg 95.1%

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

      6. metadata-eval 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. +-commutative 95.1%

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

      9. cancel-sign-sub-inv 95.1%

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

      10. div-inv 95.1%

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

      11. fabs-sub 95.1%

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

      12. add-sqr-sqrt 42.3%

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

      13. fabs-sqr 42.3%

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

      14. add-sqr-sqrt 43.7%

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

      15. associate-*l/ 47.4%

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

      16. sub-div 47.4%

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in x around 0 47.4%

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

   if 2.2499999999999999e-4 < x

   1. Initial program 89.3%

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

      1. fabs-sub 89.3%

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

      2. associate-*l/ 83.2%

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

      3. associate-*r/ 91.9%

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

      4. fmm-def 93.2%

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

      5. distribute-neg-frac 93.2%

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

      6. +-commutative 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. unsub-neg 93.2%

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

      9. metadata-eval 93.2%

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

   3. Simplified 93.2%

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

      1. fma-undefine 91.9%

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

      2. associate-*r/ 83.2%

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

      3. associate-*l/ 89.3%

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

      4. div-inv 89.1%

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

      5. sub-neg 89.1%

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

      6. metadata-eval 89.1%

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

      7. distribute-neg-in 89.1%

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

      8. +-commutative 89.1%

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

      9. cancel-sign-sub-inv 89.1%

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

      10. div-inv 89.3%

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

      11. fabs-sub 89.3%

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

      12. add-sqr-sqrt 47.6%

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

      13. fabs-sqr 47.6%

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

      14. add-sqr-sqrt 48.3%

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

      15. sub-neg 48.3%

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

      16. distribute-rgt-neg-in 48.3%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in x around inf 48.3%

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

      1. distribute-rgt-neg-out 48.3%

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

      2. distribute-lft-neg-in 48.3%

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

      3. add-sqr-sqrt 16.1%

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

      4. sqrt-unprod 31.4%

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

      5. sqr-neg 31.4%

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

      6. sqrt-unprod 20.2%

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

      7. add-sqr-sqrt 41.7%

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

      8. cancel-sign-sub-inv 41.7%

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

      9. *-un-lft-identity 41.7%

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

      10. *-commutative 41.7%

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

      11. distribute-rgt-out-- 49.7%

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

      12. add-sqr-sqrt 28.2%

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

      13. sqrt-unprod 44.7%

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

      14. sqr-neg 44.7%

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

      15. sqrt-unprod 21.4%

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

      16. add-sqr-sqrt 53.6%

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

   9. Applied egg-rr 53.6%

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

Alternative 9: 44.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\frac{y}{-1 + z}}\\ \mathbf{elif}\;x \leq 0.000205:\\ \;\;\;\;\frac{x}{y} + \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.9e-11)
   (/ x (/ y (+ -1.0 z)))
   (if (<= x 0.000205) (+ (/ x y) (/ 4.0 y)) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e-11) {
		tmp = x / (y / (-1.0 + z));
	} else if (x <= 0.000205) {
		tmp = (x / y) + (4.0 / y);
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	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 <= (-2.9d-11)) then
        tmp = x / (y / ((-1.0d0) + z))
    else if (x <= 0.000205d0) then
        tmp = (x / y) + (4.0d0 / y)
    else
        tmp = (x / y) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e-11) {
		tmp = x / (y / (-1.0 + z));
	} else if (x <= 0.000205) {
		tmp = (x / y) + (4.0 / y);
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.9e-11:
		tmp = x / (y / (-1.0 + z))
	elif x <= 0.000205:
		tmp = (x / y) + (4.0 / y)
	else:
		tmp = (x / y) * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.9e-11)
		tmp = Float64(x / Float64(y / Float64(-1.0 + z)));
	elseif (x <= 0.000205)
		tmp = Float64(Float64(x / y) + Float64(4.0 / y));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.9e-11)
		tmp = x / (y / (-1.0 + z));
	elseif (x <= 0.000205)
		tmp = (x / y) + (4.0 / y);
	else
		tmp = (x / y) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.9e-11], N[(x / N[(y / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000205], N[(N[(x / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-11

   1. Initial program 88.1%

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

      1. fabs-sub 88.1%

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

      2. associate-*l/ 81.8%

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

      3. associate-*r/ 88.2%

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

      4. fmm-def 91.5%

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

      5. distribute-neg-frac 91.5%

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

      6. +-commutative 91.5%

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

      7. distribute-neg-in 91.5%

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

      8. unsub-neg 91.5%

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

      9. metadata-eval 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. div-sub 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l/ 93.5%

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

      4. associate-/l* 99.9%

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

      5. sub-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. neg-mul-1 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-lft-in 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-lft-in 99.9%

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

      13. metadata-eval 99.9%

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

      14. neg-mul-1 99.9%

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

      15. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

      1. add-sqr-sqrt 61.4%

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

      2. fabs-sqr 61.4%

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

      3. add-sqr-sqrt 61.9%

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

      4. +-commutative 61.9%

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

      5. *-commutative 61.9%

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

      6. associate-*l/ 55.5%

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

      7. associate-*r/ 61.6%

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

      8. clear-num 61.6%

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

      9. un-div-inv 61.8%

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

      10. +-commutative 61.8%

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

   9. Applied egg-rr 61.8%

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

   if -2.9e-11 < x < 2.05e-4

   1. Initial program 95.0%

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

      1. fabs-sub 95.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 92.2%

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

      4. fmm-def 92.2%

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

      5. distribute-neg-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. distribute-neg-in 92.2%

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

      8. unsub-neg 92.2%

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

      9. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

      1. fma-undefine 92.2%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.0%

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

      4. div-inv 95.0%

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

      5. sub-neg 95.0%

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

      6. metadata-eval 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. +-commutative 95.0%

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

      9. cancel-sign-sub-inv 95.0%

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

      10. div-inv 95.0%

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

      11. fabs-sub 95.0%

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

      12. add-sqr-sqrt 41.3%

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

      13. fabs-sqr 41.3%

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

      14. add-sqr-sqrt 42.7%

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

      15. associate-*l/ 46.5%

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

      16. sub-div 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in z around 0 38.3%

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

   8. Taylor expanded in x around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. metadata-eval 38.3%

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

      3. +-commutative 38.3%

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

   10. Simplified 38.3%

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

   if 2.05e-4 < x

   1. Initial program 89.3%

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

      1. fabs-sub 89.3%

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

      2. associate-*l/ 83.2%

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

      3. associate-*r/ 91.9%

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

      4. fmm-def 93.2%

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

      5. distribute-neg-frac 93.2%

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

      6. +-commutative 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. unsub-neg 93.2%

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

      9. metadata-eval 93.2%

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

   3. Simplified 93.2%

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

      1. fma-undefine 91.9%

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

      2. associate-*r/ 83.2%

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

      3. associate-*l/ 89.3%

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

      4. div-inv 89.1%

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

      5. sub-neg 89.1%

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

      6. metadata-eval 89.1%

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

      7. distribute-neg-in 89.1%

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

      8. +-commutative 89.1%

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

      9. cancel-sign-sub-inv 89.1%

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

      10. div-inv 89.3%

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

      11. fabs-sub 89.3%

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

      12. add-sqr-sqrt 47.6%

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

      13. fabs-sqr 47.6%

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

      14. add-sqr-sqrt 48.3%

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

      15. sub-neg 48.3%

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

      16. distribute-rgt-neg-in 48.3%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in x around inf 48.3%

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

      1. distribute-rgt-neg-out 48.3%

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

      2. distribute-lft-neg-in 48.3%

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

      3. add-sqr-sqrt 16.1%

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

      4. sqrt-unprod 31.4%

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

      5. sqr-neg 31.4%

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

      6. sqrt-unprod 20.2%

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

      7. add-sqr-sqrt 41.7%

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

      8. cancel-sign-sub-inv 41.7%

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

      9. *-un-lft-identity 41.7%

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

      10. *-commutative 41.7%

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

      11. distribute-rgt-out-- 49.7%

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

      12. add-sqr-sqrt 28.2%

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

      13. sqrt-unprod 44.7%

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

      14. sqr-neg 44.7%

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

      15. sqrt-unprod 21.4%

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

      16. add-sqr-sqrt 53.6%

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

   9. Applied egg-rr 53.6%

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

Alternative 10: 44.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.000225:\\ \;\;\;\;\frac{x}{y} + \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e-11)
   (* x (/ (+ -1.0 z) y))
   (if (<= x 0.000225) (+ (/ x y) (/ 4.0 y)) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-11) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 0.000225) {
		tmp = (x / y) + (4.0 / y);
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	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 <= (-3.4d-11)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= 0.000225d0) then
        tmp = (x / y) + (4.0d0 / y)
    else
        tmp = (x / y) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-11) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 0.000225) {
		tmp = (x / y) + (4.0 / y);
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e-11:
		tmp = x * ((-1.0 + z) / y)
	elif x <= 0.000225:
		tmp = (x / y) + (4.0 / y)
	else:
		tmp = (x / y) * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e-11)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= 0.000225)
		tmp = Float64(Float64(x / y) + Float64(4.0 / y));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.4e-11)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= 0.000225)
		tmp = (x / y) + (4.0 / y);
	else
		tmp = (x / y) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e-11], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000225], N[(N[(x / y), $MachinePrecision] + N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3999999999999999e-11

   1. Initial program 88.1%

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

      1. fabs-sub 88.1%

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

      2. associate-*l/ 81.8%

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

      3. associate-*r/ 88.2%

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

      4. fmm-def 91.5%

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

      5. distribute-neg-frac 91.5%

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

      6. +-commutative 91.5%

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

      7. distribute-neg-in 91.5%

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

      8. unsub-neg 91.5%

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

      9. metadata-eval 91.5%

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

   3. Simplified 91.5%

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

      1. add-sqr-sqrt 54.7%

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

      2. fabs-sqr 54.7%

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

      3. add-sqr-sqrt 55.1%

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

      4. fma-undefine 53.4%

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

      5. associate-*r/ 47.1%

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

      6. associate-*l/ 53.5%

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

      7. div-inv 53.3%

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

      8. sub-neg 53.3%

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

      9. metadata-eval 53.3%

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

      10. distribute-neg-in 53.3%

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

      11. +-commutative 53.3%

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

      12. cancel-sign-sub-inv 53.3%

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

      13. div-inv 53.5%

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

      14. associate-*l/ 47.1%

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

      15. sub-div 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in x around inf 55.5%

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

      1. associate-/l* 61.6%

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

      2. sub-neg 61.6%

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

      3. metadata-eval 61.6%

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

   9. Simplified 61.6%

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

   if -3.3999999999999999e-11 < x < 2.2499999999999999e-4

   1. Initial program 95.0%

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

      1. fabs-sub 95.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 92.2%

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

      4. fmm-def 92.2%

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

      5. distribute-neg-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. distribute-neg-in 92.2%

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

      8. unsub-neg 92.2%

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

      9. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

      1. fma-undefine 92.2%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.0%

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

      4. div-inv 95.0%

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

      5. sub-neg 95.0%

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

      6. metadata-eval 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. +-commutative 95.0%

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

      9. cancel-sign-sub-inv 95.0%

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

      10. div-inv 95.0%

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

      11. fabs-sub 95.0%

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

      12. add-sqr-sqrt 41.3%

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

      13. fabs-sqr 41.3%

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

      14. add-sqr-sqrt 42.7%

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

      15. associate-*l/ 46.5%

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

      16. sub-div 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in z around 0 38.3%

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

   8. Taylor expanded in x around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. metadata-eval 38.3%

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

      3. +-commutative 38.3%

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

   10. Simplified 38.3%

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

   if 2.2499999999999999e-4 < x

   1. Initial program 89.3%

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

      1. fabs-sub 89.3%

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

      2. associate-*l/ 83.2%

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

      3. associate-*r/ 91.9%

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

      4. fmm-def 93.2%

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

      5. distribute-neg-frac 93.2%

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

      6. +-commutative 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. unsub-neg 93.2%

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

      9. metadata-eval 93.2%

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

   3. Simplified 93.2%

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

      1. fma-undefine 91.9%

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

      2. associate-*r/ 83.2%

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

      3. associate-*l/ 89.3%

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

      4. div-inv 89.1%

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

      5. sub-neg 89.1%

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

      6. metadata-eval 89.1%

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

      7. distribute-neg-in 89.1%

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

      8. +-commutative 89.1%

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

      9. cancel-sign-sub-inv 89.1%

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

      10. div-inv 89.3%

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

      11. fabs-sub 89.3%

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

      12. add-sqr-sqrt 47.6%

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

      13. fabs-sqr 47.6%

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

      14. add-sqr-sqrt 48.3%

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

      15. sub-neg 48.3%

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

      16. distribute-rgt-neg-in 48.3%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in x around inf 48.3%

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

      1. distribute-rgt-neg-out 48.3%

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

      2. distribute-lft-neg-in 48.3%

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

      3. add-sqr-sqrt 16.1%

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

      4. sqrt-unprod 31.4%

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

      5. sqr-neg 31.4%

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

      6. sqrt-unprod 20.2%

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

      7. add-sqr-sqrt 41.7%

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

      8. cancel-sign-sub-inv 41.7%

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

      9. *-un-lft-identity 41.7%

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

      10. *-commutative 41.7%

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

      11. distribute-rgt-out-- 49.7%

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

      12. add-sqr-sqrt 28.2%

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

      13. sqrt-unprod 44.7%

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

      14. sqr-neg 44.7%

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

      15. sqrt-unprod 21.4%

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

      16. add-sqr-sqrt 53.6%

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

   9. Applied egg-rr 53.6%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.000225:\\ \;\;\;\;\frac{x}{y} + \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.000185:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e-11)
   (* x (/ (+ -1.0 z) y))
   (if (<= x 0.000185) (/ (+ x 4.0) y) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-11) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 0.000185) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	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 <= (-3.4d-11)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= 0.000185d0) then
        tmp = (x + 4.0d0) / y
    else
        tmp = (x / y) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-11) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 0.000185) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e-11:
		tmp = x * ((-1.0 + z) / y)
	elif x <= 0.000185:
		tmp = (x + 4.0) / y
	else:
		tmp = (x / y) * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e-11)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= 0.000185)
		tmp = Float64(Float64(x + 4.0) / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.4e-11)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= 0.000185)
		tmp = (x + 4.0) / y;
	else
		tmp = (x / y) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e-11], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000185], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3999999999999999e-11

   1. Initial program 88.1%

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

      1. fabs-sub 88.1%

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

      2. associate-*l/ 81.8%

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

      3. associate-*r/ 88.2%

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

      4. fmm-def 91.5%

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

      5. distribute-neg-frac 91.5%

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

      6. +-commutative 91.5%

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

      7. distribute-neg-in 91.5%

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

      8. unsub-neg 91.5%

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

      9. metadata-eval 91.5%

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

   3. Simplified 91.5%

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

      1. add-sqr-sqrt 54.7%

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

      2. fabs-sqr 54.7%

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

      3. add-sqr-sqrt 55.1%

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

      4. fma-undefine 53.4%

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

      5. associate-*r/ 47.1%

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

      6. associate-*l/ 53.5%

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

      7. div-inv 53.3%

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

      8. sub-neg 53.3%

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

      9. metadata-eval 53.3%

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

      10. distribute-neg-in 53.3%

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

      11. +-commutative 53.3%

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

      12. cancel-sign-sub-inv 53.3%

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

      13. div-inv 53.5%

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

      14. associate-*l/ 47.1%

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

      15. sub-div 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in x around inf 55.5%

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

      1. associate-/l* 61.6%

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

      2. sub-neg 61.6%

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

      3. metadata-eval 61.6%

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

   9. Simplified 61.6%

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

   if -3.3999999999999999e-11 < x < 1.85e-4

   1. Initial program 95.0%

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

      1. fabs-sub 95.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 92.2%

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

      4. fmm-def 92.2%

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

      5. distribute-neg-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. distribute-neg-in 92.2%

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

      8. unsub-neg 92.2%

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

      9. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

      1. fma-undefine 92.2%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.0%

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

      4. div-inv 95.0%

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

      5. sub-neg 95.0%

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

      6. metadata-eval 95.0%

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

      7. distribute-neg-in 95.0%

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

      8. +-commutative 95.0%

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

      9. cancel-sign-sub-inv 95.0%

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

      10. div-inv 95.0%

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

      11. fabs-sub 95.0%

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

      12. add-sqr-sqrt 41.3%

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

      13. fabs-sqr 41.3%

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

      14. add-sqr-sqrt 42.7%

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

      15. associate-*l/ 46.5%

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

      16. sub-div 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in z around 0 38.3%

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

   if 1.85e-4 < x

   1. Initial program 89.3%

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

      1. fabs-sub 89.3%

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

      2. associate-*l/ 83.2%

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

      3. associate-*r/ 91.9%

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

      4. fmm-def 93.2%

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

      5. distribute-neg-frac 93.2%

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

      6. +-commutative 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. unsub-neg 93.2%

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

      9. metadata-eval 93.2%

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

   3. Simplified 93.2%

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

      1. fma-undefine 91.9%

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

      2. associate-*r/ 83.2%

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

      3. associate-*l/ 89.3%

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

      4. div-inv 89.1%

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

      5. sub-neg 89.1%

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

      6. metadata-eval 89.1%

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

      7. distribute-neg-in 89.1%

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

      8. +-commutative 89.1%

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

      9. cancel-sign-sub-inv 89.1%

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

      10. div-inv 89.3%

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

      11. fabs-sub 89.3%

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

      12. add-sqr-sqrt 47.6%

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

      13. fabs-sqr 47.6%

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

      14. add-sqr-sqrt 48.3%

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

      15. sub-neg 48.3%

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

      16. distribute-rgt-neg-in 48.3%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in x around inf 48.3%

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

      1. distribute-rgt-neg-out 48.3%

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

      2. distribute-lft-neg-in 48.3%

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

      3. add-sqr-sqrt 16.1%

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

      4. sqrt-unprod 31.4%

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

      5. sqr-neg 31.4%

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

      6. sqrt-unprod 20.2%

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

      7. add-sqr-sqrt 41.7%

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

      8. cancel-sign-sub-inv 41.7%

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

      9. *-un-lft-identity 41.7%

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

      10. *-commutative 41.7%

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

      11. distribute-rgt-out-- 49.7%

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

      12. add-sqr-sqrt 28.2%

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

      13. sqrt-unprod 44.7%

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

      14. sqr-neg 44.7%

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

      15. sqrt-unprod 21.4%

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

      16. add-sqr-sqrt 53.6%

          \[\leadsto \frac{x}{y} \cdot \left(1 - \color{blue}{z}\right) \]

   9. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(1 - z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 0.000185:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-1}{y} \cdot x\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55) (* (/ -1.0 y) x) (if (<= x 4.0) (/ 4.0 y) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55) {
		tmp = (-1.0 / y) * x;
	} else if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = ((-1.0d0) / y) * x
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55) {
		tmp = (-1.0 / y) * x;
	} else if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55:
		tmp = (-1.0 / y) * x
	elif x <= 4.0:
		tmp = 4.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(-1.0 / y) * x);
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = (-1.0 / y) * x;
	elseif (x <= 4.0)
		tmp = 4.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55], N[(N[(-1.0 / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 87.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 87.4%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 80.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 90.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.8%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 56.8%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 57.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 55.4%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 48.7%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 55.5%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 55.3%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 55.3%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 55.3%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 55.5%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 48.7%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 57.6%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 57.6%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in x around inf 57.7%

      \[\leadsto \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   8. Step-by-step derivation

      1. associate-/l* 64.2%

         \[\leadsto \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 64.2%

         \[\leadsto x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

      3. metadata-eval 64.2%

         \[\leadsto x \cdot \frac{z + \color{blue}{-1}}{y} \]

   9. Simplified 64.2%

      \[\leadsto \color{blue}{x \cdot \frac{z + -1}{y}} \]

   10. Taylor expanded in z around 0 35.5%

       \[\leadsto x \cdot \color{blue}{\frac{-1}{y}} \]

   if -1.55000000000000004 < x < 4

   1. Initial program 95.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 95.2%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 99.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.6%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 92.5%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 99.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 95.2%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 95.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 95.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 95.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 95.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 95.2%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 95.2%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 95.2%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 95.2%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 41.7%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 41.7%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 43.0%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 46.6%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 46.6%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around 0 36.7%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 89.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 89.0%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 82.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 91.7%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 93.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 91.7%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 82.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 89.0%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 88.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 89.0%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 89.0%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 48.9%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 48.9%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 49.6%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. sub-neg 49.6%

          \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      16. distribute-rgt-neg-in 49.6%

          \[\leadsto \frac{x + 4}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)} \]

   6. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{x + 4}{y} + \frac{x}{y} \cdot \left(-z\right)} \]

   7. Taylor expanded in x around inf 49.6%

      \[\leadsto \color{blue}{\frac{x}{y}} + \frac{x}{y} \cdot \left(-z\right) \]

   8. Taylor expanded in z around 0 37.1%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-1}{y} \cdot x\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.9% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e-11) (* x (/ (+ -1.0 z) y)) (/ (+ x 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e-11) {
		tmp = x * ((-1.0 + 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 <= (-3.4d-11)) then
        tmp = x * (((-1.0d0) + 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 <= -3.4e-11) {
		tmp = x * ((-1.0 + z) / y);
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e-11:
		tmp = x * ((-1.0 + z) / y)
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e-11)
		tmp = Float64(x * Float64(Float64(-1.0 + 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 <= -3.4e-11)
		tmp = x * ((-1.0 + z) / y);
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e-11], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3999999999999999e-11

   1. Initial program 88.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 88.1%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 81.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 88.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 91.5%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 91.5%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.7%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 54.7%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 55.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 53.4%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 47.1%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 53.5%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 53.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 53.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 53.3%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 53.3%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 53.3%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 53.3%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 53.5%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 47.1%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 55.5%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in x around inf 55.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   8. Step-by-step derivation

      1. associate-/l* 61.6%

         \[\leadsto \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 61.6%

         \[\leadsto x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

      3. metadata-eval 61.6%

         \[\leadsto x \cdot \frac{z + \color{blue}{-1}}{y} \]

   9. Simplified 61.6%

      \[\leadsto \color{blue}{x \cdot \frac{z + -1}{y}} \]

   if -3.3999999999999999e-11 < x

   1. Initial program 92.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.8%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 93.5%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.1%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.6%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.6%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 92.1%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 93.5%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 92.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 92.8%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 92.8%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 43.7%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 43.7%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 44.8%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 45.7%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 47.7%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 37.5%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.5% 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 87.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 87.4%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 80.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 90.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.8%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 56.8%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 57.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 55.4%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 48.7%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 55.5%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 55.3%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 55.3%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 55.3%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 55.5%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 48.7%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 57.6%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 57.6%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in z around 0 35.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{4 + x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 35.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(4 + x\right)}{y}} \]

      2. distribute-lft-in 35.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 4 + -1 \cdot x}}{y} \]

      3. metadata-eval 35.7%

         \[\leadsto \frac{\color{blue}{-4} + -1 \cdot x}{y} \]

      4. neg-mul-1 35.7%

         \[\leadsto \frac{-4 + \color{blue}{\left(-x\right)}}{y} \]

      5. sub-neg 35.7%

         \[\leadsto \frac{\color{blue}{-4 - x}}{y} \]

   9. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{-4 - x}{y}} \]

   if -4 < x

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.9%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 93.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.7%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.7%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 92.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 93.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 92.9%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.9%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 92.9%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 92.9%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 44.3%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 44.3%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 45.4%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 46.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 48.3%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 36.9%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-4 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-1}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.0) (* (/ -1.0 y) x) (/ (+ x 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.0) {
		tmp = (-1.0 / y) * x;
	} 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 = ((-1.0d0) / y) * x
    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 = (-1.0 / y) * x;
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.0:
		tmp = (-1.0 / y) * x
	else:
		tmp = (x + 4.0) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(Float64(-1.0 / y) * x);
	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 = (-1.0 / y) * x;
	else
		tmp = (x + 4.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.0], N[(N[(-1.0 / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 87.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 87.4%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 80.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 87.4%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 90.9%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 90.9%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 56.8%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

      2. fabs-sqr 56.8%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

      3. add-sqr-sqrt 57.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

      4. fma-undefine 55.4%

         \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

      5. associate-*r/ 48.7%

         \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

      6. associate-*l/ 55.5%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

      7. div-inv 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

      8. sub-neg 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

      9. metadata-eval 55.3%

         \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

      10. distribute-neg-in 55.3%

          \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

      11. +-commutative 55.3%

          \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

      12. cancel-sign-sub-inv 55.3%

          \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

      13. div-inv 55.5%

          \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

      14. associate-*l/ 48.7%

          \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

      15. sub-div 57.6%

          \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   6. Applied egg-rr 57.6%

      \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

   7. Taylor expanded in x around inf 57.7%

      \[\leadsto \color{blue}{\frac{x \cdot \left(z - 1\right)}{y}} \]
   8. Step-by-step derivation

      1. associate-/l* 64.2%

         \[\leadsto \color{blue}{x \cdot \frac{z - 1}{y}} \]

      2. sub-neg 64.2%

         \[\leadsto x \cdot \frac{\color{blue}{z + \left(-1\right)}}{y} \]

      3. metadata-eval 64.2%

         \[\leadsto x \cdot \frac{z + \color{blue}{-1}}{y} \]

   9. Simplified 64.2%

      \[\leadsto \color{blue}{x \cdot \frac{z + -1}{y}} \]

   10. Taylor expanded in z around 0 35.5%

       \[\leadsto x \cdot \color{blue}{\frac{-1}{y}} \]

   if -4 < x

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.9%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 93.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 92.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.7%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.7%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.7%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 92.2%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 93.6%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 92.9%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.9%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.9%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 92.9%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 92.9%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 44.3%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 44.3%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 45.4%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 46.3%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 48.3%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in z around 0 36.9%

      \[\leadsto \color{blue}{\frac{4 + x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-1}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.7% 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 92.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 92.8%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 94.0%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 91.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 92.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 92.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 92.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 92.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 92.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 92.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 92.0%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 91.0%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 94.0%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 92.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 92.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 92.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 92.8%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 92.8%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 38.7%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 38.7%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 39.8%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. associate-*l/ 42.4%

          \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

      16. sub-div 43.4%

          \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   6. Applied egg-rr 43.4%

      \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

   7. Taylor expanded in x around 0 26.5%

      \[\leadsto \color{blue}{\frac{4}{y}} \]

   if 4 < x

   1. Initial program 89.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. fabs-sub 89.0%

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/ 82.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/ 91.7%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fmm-def 93.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval 93.0%

         \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 91.7%

         \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

      2. associate-*r/ 82.8%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

      3. associate-*l/ 89.0%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

      4. div-inv 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

      5. sub-neg 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

      6. metadata-eval 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

      7. distribute-neg-in 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

      8. +-commutative 88.8%

         \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

      9. cancel-sign-sub-inv 88.8%

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

      10. div-inv 89.0%

          \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

      11. fabs-sub 89.0%

          \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      12. add-sqr-sqrt 48.9%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

      13. fabs-sqr 48.9%

          \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

      14. add-sqr-sqrt 49.6%

          \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

      15. sub-neg 49.6%

          \[\leadsto \color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)} \]

      16. distribute-rgt-neg-in 49.6%

          \[\leadsto \frac{x + 4}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)} \]

   6. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{x + 4}{y} + \frac{x}{y} \cdot \left(-z\right)} \]

   7. Taylor expanded in x around inf 49.6%

      \[\leadsto \color{blue}{\frac{x}{y}} + \frac{x}{y} \cdot \left(-z\right) \]

   8. Taylor expanded in z around 0 37.1%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 20.6% 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 91.7%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 91.7%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 90.8%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 91.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 92.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 92.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 91.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}}\right| \]

   2. associate-*r/ 90.8%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y}\right| \]

   3. associate-*l/ 91.7%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y}\right| \]

   4. div-inv 91.6%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}}\right| \]

   5. sub-neg 91.6%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y}\right| \]

   6. metadata-eval 91.6%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y}\right| \]

   7. distribute-neg-in 91.6%

      \[\leadsto \left|\frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y}\right| \]

   8. +-commutative 91.6%

      \[\leadsto \left|\frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y}\right| \]

   9. cancel-sign-sub-inv 91.6%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}}\right| \]

   10. div-inv 91.7%

       \[\leadsto \left|\frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}}\right| \]

   11. fabs-sub 91.7%

       \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

   12. add-sqr-sqrt 41.6%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right| \]

   13. fabs-sqr 41.6%

       \[\leadsto \color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}} \]

   14. add-sqr-sqrt 42.6%

       \[\leadsto \color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \]

   15. associate-*l/ 43.3%

       \[\leadsto \frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}} \]

   16. sub-div 45.6%

       \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

6. Applied egg-rr 45.6%

   \[\leadsto \color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}} \]

7. Taylor expanded in x around 0 19.8%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Alternative 18: 20.7% 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 91.7%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. fabs-sub 91.7%

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   2. associate-*l/ 90.8%

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   3. associate-*r/ 91.2%

      \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

   4. fmm-def 92.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

   5. distribute-neg-frac 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

   6. +-commutative 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

   7. distribute-neg-in 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

   8. unsub-neg 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

   9. metadata-eval 92.3%

      \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Simplified 92.3%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 50.2%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}}\right| \]

   2. fabs-sqr 50.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)}} \]

   3. add-sqr-sqrt 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)} \]

   4. fma-undefine 50.2%

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + \frac{-4 - x}{y}} \]

   5. associate-*r/ 48.8%

      \[\leadsto \color{blue}{\frac{x \cdot z}{y}} + \frac{-4 - x}{y} \]

   6. associate-*l/ 50.4%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot z} + \frac{-4 - x}{y} \]

   7. div-inv 50.4%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 - x\right) \cdot \frac{1}{y}} \]

   8. sub-neg 50.4%

      \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-4 + \left(-x\right)\right)} \cdot \frac{1}{y} \]

   9. metadata-eval 50.4%

      \[\leadsto \frac{x}{y} \cdot z + \left(\color{blue}{\left(-4\right)} + \left(-x\right)\right) \cdot \frac{1}{y} \]

   10. distribute-neg-in 50.4%

       \[\leadsto \frac{x}{y} \cdot z + \color{blue}{\left(-\left(4 + x\right)\right)} \cdot \frac{1}{y} \]

   11. +-commutative 50.4%

       \[\leadsto \frac{x}{y} \cdot z + \left(-\color{blue}{\left(x + 4\right)}\right) \cdot \frac{1}{y} \]

   12. cancel-sign-sub-inv 50.4%

       \[\leadsto \color{blue}{\frac{x}{y} \cdot z - \left(x + 4\right) \cdot \frac{1}{y}} \]

   13. div-inv 50.4%

       \[\leadsto \frac{x}{y} \cdot z - \color{blue}{\frac{x + 4}{y}} \]

   14. associate-*l/ 48.8%

       \[\leadsto \color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y} \]

   15. sub-div 51.5%

       \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

6. Applied egg-rr 51.5%

   \[\leadsto \color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}} \]

7. Taylor expanded in x around 0 22.2%

   \[\leadsto \color{blue}{\frac{-4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))