fabs fraction 1

Percentage Accurate: 91.5% → 99.4%

Time: 11.1s

Alternatives: 14

Speedup: 0.5×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.5% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-76} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+283}\right):\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* z (/ x y))))))
   (if (or (<= t_0 4e-76) (not (<= t_0 5e+283)))
     (fabs (* (/ -1.0 y) (fma x z (- -4.0 x))))
     t_0)))

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 <= 4e-76) || !(t_0 <= 5e+283)) {
		tmp = fabs(((-1.0 / y) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = t_0;
	}
	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 <= 4e-76) || !(t_0 <= 5e+283))
		tmp = abs(Float64(Float64(-1.0 / y) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = t_0;
	end
	return 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[Or[LessEqual[t$95$0, 4e-76], N[Not[LessEqual[t$95$0, 5e+283]], $MachinePrecision]], N[Abs[N[(N[(-1.0 / y), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]

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))) < 3.99999999999999971e-76 or 5.0000000000000004e283 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 76.7%

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

   3. Simplified 99.9%

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

   if 3.99999999999999971e-76 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 5.0000000000000004e283

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 4 \cdot 10^{-76} \lor \neg \left(\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 5 \cdot 10^{+283}\right):\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* z (/ x y))))))
   (if (<= t_0 INFINITY) t_0 (fabs (* (/ -1.0 y) (* x 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 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fabs(((-1.0 / y) * (x * z)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((((x + 4.0) / y) - (z * (x / y))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((-1.0 / y) * (x * 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 <= math.inf:
		tmp = t_0
	else:
		tmp = math.fabs(((-1.0 / y) * (x * 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 <= Inf)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(-1.0 / y) * Float64(x * z)));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 56.9%

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

4. Final simplification 93.4%

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

Alternative 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-89}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-89)
   (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 <= 8e-89) {
		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 <= 8e-89)
		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, 8e-89], 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 < 8.00000000000000031e-89

   1. Initial program 82.7%

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

   3. Simplified 96.6%

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

   if 8.00000000000000031e-89 < y

   1. Initial program 96.5%

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

      1. fabs-sub 96.5%

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

      2. associate-*l/ 95.2%

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

      3. associate-*r/ 98.7%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

Alternative 4: 46.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq -0.52:\\ \;\;\;\;x \cdot \frac{z}{y} - \frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e+130)
   (/ (- x) y)
   (if (<= x -0.52)
     (- (* x (/ z y)) (/ (+ x 4.0) y))
     (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+130) {
		tmp = -x / y;
	} else if (x <= -0.52) {
		tmp = (x * (z / y)) - ((x + 4.0) / y);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.55d+130)) then
        tmp = -x / y
    else if (x <= (-0.52d0)) then
        tmp = (x * (z / y)) - ((x + 4.0d0) / y)
    else
        tmp = ((x + 4.0d0) - (x * z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+130) {
		tmp = -x / y;
	} else if (x <= -0.52) {
		tmp = (x * (z / y)) - ((x + 4.0) / y);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55e+130:
		tmp = -x / y
	elif x <= -0.52:
		tmp = (x * (z / y)) - ((x + 4.0) / y)
	else:
		tmp = ((x + 4.0) - (x * z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e+130)
		tmp = Float64(Float64(-x) / y);
	elseif (x <= -0.52)
		tmp = Float64(Float64(x * Float64(z / y)) - Float64(Float64(x + 4.0) / y));
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.55e+130)
		tmp = -x / y;
	elseif (x <= -0.52)
		tmp = (x * (z / y)) - ((x + 4.0) / y);
	else
		tmp = ((x + 4.0) - (x * z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e+130], N[((-x) / y), $MachinePrecision], If[LessEqual[x, -0.52], N[(N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e130

   1. Initial program 67.5%

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

      1. fabs-sub 67.5%

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

      2. associate-*l/ 70.6%

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

      3. associate-*r/ 75.5%

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

      4. fmm-def 89.0%

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

      5. distribute-neg-frac 89.0%

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

      6. +-commutative 89.0%

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

      7. distribute-neg-in 89.0%

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

      8. unsub-neg 89.0%

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

      9. metadata-eval 89.0%

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

   3. Simplified 89.0%

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

      1. add-sqr-sqrt 56.4%

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

      2. fabs-sqr 56.4%

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

      3. add-sqr-sqrt 56.8%

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

      4. fma-undefine 43.3%

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

      5. associate-*r/ 40.8%

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

      6. associate-*l/ 37.9%

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

      7. div-inv 37.9%

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

      8. sub-neg 37.9%

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

      9. metadata-eval 37.9%

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

      10. distribute-neg-in 37.9%

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

      11. +-commutative 37.9%

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

      12. cancel-sign-sub-inv 37.9%

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

      13. div-inv 37.9%

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

      14. associate-*l/ 40.8%

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

      15. sub-div 54.3%

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

   6. Applied egg-rr 54.3%

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

   7. Taylor expanded in z around 0 41.4%

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

      1. associate-*r/ 41.4%

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

      2. distribute-lft-in 41.4%

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

      3. metadata-eval 41.4%

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

      4. neg-mul-1 41.4%

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

      5. sub-neg 41.4%

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

   9. Simplified 41.4%

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

   10. Taylor expanded in x around inf 41.4%

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

       1. neg-mul-1 41.4%

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

   12. Simplified 41.4%

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

   if -1.55e130 < x < -0.52000000000000002

   1. Initial program 92.2%

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

      1. fabs-sub 92.2%

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

      2. associate-*l/ 96.2%

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

      3. associate-*r/ 99.9%

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

      4. fmm-def 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. unsub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 45.9%

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

      2. fabs-sqr 45.9%

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

      3. add-sqr-sqrt 47.0%

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

      4. fma-undefine 47.0%

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

      5. associate-*r/ 43.3%

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

      6. associate-*l/ 39.3%

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

      7. div-inv 39.3%

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

      8. sub-neg 39.3%

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

      9. metadata-eval 39.3%

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

      10. distribute-neg-in 39.3%

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

      11. +-commutative 39.3%

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

      12. cancel-sign-sub-inv 39.3%

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

      13. associate-*l/ 43.3%

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

      14. associate-*r/ 47.0%

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

      15. div-inv 47.0%

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

   6. Applied egg-rr 47.0%

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

   if -0.52000000000000002 < x

   1. Initial program 90.2%

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

      1. fabs-sub 90.2%

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

      2. associate-*l/ 93.9%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. fma-undefine 92.9%

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

      2. associate-*r/ 93.9%

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

      3. associate-*l/ 90.2%

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

      4. div-inv 90.1%

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

      5. sub-neg 90.1%

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

      6. metadata-eval 90.1%

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

      7. distribute-neg-in 90.1%

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

      8. +-commutative 90.1%

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

      9. cancel-sign-sub-inv 90.1%

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

      10. div-inv 90.2%

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

      11. fabs-sub 90.2%

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

      12. add-sqr-sqrt 46.7%

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

      13. fabs-sqr 46.7%

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

      14. add-sqr-sqrt 47.7%

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

      15. associate-*l/ 50.5%

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

      16. sub-div 52.0%

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

   6. Applied egg-rr 52.0%

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

Alternative 5: 49.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.39)
   (* x (/ (+ -1.0 z) y))
   (if (<= x 2.5e-8) (/ (- 4.0 (* x z)) y) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.39) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 2.5e-8) {
		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 <= (-0.39d0)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= 2.5d-8) 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 <= -0.39) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 2.5e-8) {
		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 <= -0.39:
		tmp = x * ((-1.0 + z) / y)
	elif x <= 2.5e-8:
		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 <= -0.39)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= 2.5e-8)
		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 <= -0.39)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= 2.5e-8)
		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, -0.39], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-8], 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 < -0.39000000000000001

   1. Initial program 77.7%

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

      1. fabs-sub 77.7%

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

      2. associate-*l/ 81.1%

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

      3. associate-*r/ 85.6%

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

      4. fmm-def 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. +-commutative 93.5%

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

      7. distribute-neg-in 93.5%

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

      8. unsub-neg 93.5%

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

      9. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

      1. add-sqr-sqrt 52.1%

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

      2. fabs-sqr 52.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. fma-undefine 44.8%

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

      5. associate-*r/ 41.9%

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

      6. associate-*l/ 38.5%

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

      7. div-inv 38.5%

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

      8. sub-neg 38.5%

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

      9. metadata-eval 38.5%

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

      10. distribute-neg-in 38.5%

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

      11. +-commutative 38.5%

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

      12. cancel-sign-sub-inv 38.5%

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

      13. div-inv 38.5%

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

      14. associate-*l/ 41.9%

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

      15. sub-div 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/l* 52.7%

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

      2. sub-neg 52.7%

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

      3. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   if -0.39000000000000001 < x < 2.4999999999999999e-8

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 93.6%

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

      4. fmm-def 93.6%

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

      5. distribute-neg-frac 93.6%

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

      6. +-commutative 93.6%

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

      7. distribute-neg-in 93.6%

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

      8. unsub-neg 93.6%

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

      9. metadata-eval 93.6%

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

   3. Simplified 93.6%

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

      1. fma-undefine 93.6%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 95.9%

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

      4. div-inv 95.9%

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

      5. sub-neg 95.9%

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

      6. metadata-eval 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. +-commutative 95.9%

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

      9. cancel-sign-sub-inv 95.9%

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

      10. div-inv 95.9%

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

      11. fabs-sub 95.9%

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

      12. add-sqr-sqrt 51.4%

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

      13. fabs-sqr 51.4%

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

      14. add-sqr-sqrt 52.8%

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

      15. associate-*l/ 55.6%

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

      16. sub-div 55.6%

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

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in x around 0 55.5%

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

   if 2.4999999999999999e-8 < x

   1. Initial program 81.1%

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

      1. fabs-sub 81.1%

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

      2. associate-*l/ 84.2%

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

      3. associate-*r/ 91.8%

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

      4. fmm-def 93.1%

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

      5. distribute-neg-frac 93.1%

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

      6. +-commutative 93.1%

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

      7. distribute-neg-in 93.1%

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

      8. unsub-neg 93.1%

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

      9. metadata-eval 93.1%

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

   3. Simplified 93.1%

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

      1. fma-undefine 91.8%

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

      2. associate-*r/ 84.2%

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

      3. associate-*l/ 81.1%

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

      4. div-inv 80.9%

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

      5. sub-neg 80.9%

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

      6. metadata-eval 80.9%

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

      7. distribute-neg-in 80.9%

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

      8. +-commutative 80.9%

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

      9. cancel-sign-sub-inv 80.9%

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

      10. div-inv 81.1%

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

      11. fabs-sub 81.1%

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

      12. add-sqr-sqrt 39.2%

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

      13. fabs-sqr 39.2%

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

      14. add-sqr-sqrt 39.5%

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

      15. associate-*l/ 42.2%

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

      16. sub-div 46.2%

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in x around inf 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-*r/ 47.6%

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

   9. Simplified 47.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{4 - x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.47)
   (* x (/ (+ -1.0 z) y))
   (if (<= x 2.5e-8) (/ (+ x 4.0) y) (* (/ x y) (- 1.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.47) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 2.5e-8) {
		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 <= (-0.47d0)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= 2.5d-8) 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 <= -0.47) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 2.5e-8) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = (x / y) * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.47:
		tmp = x * ((-1.0 + z) / y)
	elif x <= 2.5e-8:
		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 <= -0.47)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= 2.5e-8)
		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 <= -0.47)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= 2.5e-8)
		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, -0.47], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-8], 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 < -0.46999999999999997

   1. Initial program 77.7%

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

      1. fabs-sub 77.7%

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

      2. associate-*l/ 81.1%

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

      3. associate-*r/ 85.6%

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

      4. fmm-def 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. +-commutative 93.5%

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

      7. distribute-neg-in 93.5%

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

      8. unsub-neg 93.5%

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

      9. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

      1. add-sqr-sqrt 52.1%

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

      2. fabs-sqr 52.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. fma-undefine 44.8%

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

      5. associate-*r/ 41.9%

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

      6. associate-*l/ 38.5%

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

      7. div-inv 38.5%

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

      8. sub-neg 38.5%

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

      9. metadata-eval 38.5%

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

      10. distribute-neg-in 38.5%

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

      11. +-commutative 38.5%

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

      12. cancel-sign-sub-inv 38.5%

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

      13. div-inv 38.5%

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

      14. associate-*l/ 41.9%

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

      15. sub-div 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/l* 52.7%

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

      2. sub-neg 52.7%

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

      3. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   if -0.46999999999999997 < x < 2.4999999999999999e-8

   1. Initial program 95.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 54.2%

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

      2. fabs-sqr 54.2%

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

      3. add-sqr-sqrt 55.6%

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

      4. associate-*l/ 55.6%

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

      5. clear-num 55.6%

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

      6. mul-1-neg 55.6%

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

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in z around 0 41.6%

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

      1. +-commutative 41.6%

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

   9. Simplified 41.6%

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

   if 2.4999999999999999e-8 < x

   1. Initial program 81.1%

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

      1. fabs-sub 81.1%

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

      2. associate-*l/ 84.2%

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

      3. associate-*r/ 91.8%

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

      4. fmm-def 93.1%

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

      5. distribute-neg-frac 93.1%

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

      6. +-commutative 93.1%

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

      7. distribute-neg-in 93.1%

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

      8. unsub-neg 93.1%

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

      9. metadata-eval 93.1%

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

   3. Simplified 93.1%

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

      1. fma-undefine 91.8%

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

      2. associate-*r/ 84.2%

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

      3. associate-*l/ 81.1%

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

      4. div-inv 80.9%

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

      5. sub-neg 80.9%

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

      6. metadata-eval 80.9%

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

      7. distribute-neg-in 80.9%

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

      8. +-commutative 80.9%

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

      9. cancel-sign-sub-inv 80.9%

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

      10. div-inv 81.1%

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

      11. fabs-sub 81.1%

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

      12. add-sqr-sqrt 39.2%

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

      13. fabs-sqr 39.2%

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

      14. add-sqr-sqrt 39.5%

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

      15. associate-*l/ 42.2%

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

      16. sub-div 46.2%

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in x around inf 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-*r/ 47.6%

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

   9. Simplified 47.6%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.47:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.5)
   (* x (/ (+ -1.0 z) y))
   (if (<= x 2.5e-8) (/ (+ x 4.0) y) (* x (/ (- 1.0 z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.5) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 2.5e-8) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-0.5d0)) then
        tmp = x * (((-1.0d0) + z) / y)
    else if (x <= 2.5d-8) then
        tmp = (x + 4.0d0) / y
    else
        tmp = x * ((1.0d0 - z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.5) {
		tmp = x * ((-1.0 + z) / y);
	} else if (x <= 2.5e-8) {
		tmp = (x + 4.0) / y;
	} else {
		tmp = x * ((1.0 - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.5:
		tmp = x * ((-1.0 + z) / y)
	elif x <= 2.5e-8:
		tmp = (x + 4.0) / y
	else:
		tmp = x * ((1.0 - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.5)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	elseif (x <= 2.5e-8)
		tmp = Float64(Float64(x + 4.0) / y);
	else
		tmp = Float64(x * Float64(Float64(1.0 - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.5)
		tmp = x * ((-1.0 + z) / y);
	elseif (x <= 2.5e-8)
		tmp = (x + 4.0) / y;
	else
		tmp = x * ((1.0 - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.5], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-8], N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 77.7%

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

      1. fabs-sub 77.7%

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

      2. associate-*l/ 81.1%

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

      3. associate-*r/ 85.6%

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

      4. fmm-def 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. +-commutative 93.5%

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

      7. distribute-neg-in 93.5%

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

      8. unsub-neg 93.5%

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

      9. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

      1. add-sqr-sqrt 52.1%

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

      2. fabs-sqr 52.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. fma-undefine 44.8%

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

      5. associate-*r/ 41.9%

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

      6. associate-*l/ 38.5%

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

      7. div-inv 38.5%

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

      8. sub-neg 38.5%

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

      9. metadata-eval 38.5%

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

      10. distribute-neg-in 38.5%

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

      11. +-commutative 38.5%

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

      12. cancel-sign-sub-inv 38.5%

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

      13. div-inv 38.5%

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

      14. associate-*l/ 41.9%

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

      15. sub-div 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/l* 52.7%

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

      2. sub-neg 52.7%

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

      3. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   if -0.5 < x < 2.4999999999999999e-8

   1. Initial program 95.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 54.2%

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

      2. fabs-sqr 54.2%

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

      3. add-sqr-sqrt 55.6%

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

      4. associate-*l/ 55.6%

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

      5. clear-num 55.6%

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

      6. mul-1-neg 55.6%

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

   6. Applied egg-rr 55.6%

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

   7. Taylor expanded in z around 0 41.6%

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

      1. +-commutative 41.6%

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

   9. Simplified 41.6%

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

   if 2.4999999999999999e-8 < x

   1. Initial program 81.1%

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

   3. Simplified 92.1%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
   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 46.1%

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

      4. associate-*l/ 46.2%

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

      5. clear-num 46.2%

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

      6. mul-1-neg 46.2%

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in x around inf 46.2%

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

      1. associate-/l* 47.4%

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

   9. Simplified 47.4%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x + 4}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.0% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.52) (* x (/ (+ -1.0 z) y)) (/ (- (+ x 4.0) (* x z)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.52) {
		tmp = x * ((-1.0 + z) / y);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-0.52d0)) then
        tmp = x * (((-1.0d0) + z) / y)
    else
        tmp = ((x + 4.0d0) - (x * z)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.52) {
		tmp = x * ((-1.0 + z) / y);
	} else {
		tmp = ((x + 4.0) - (x * z)) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.52:
		tmp = x * ((-1.0 + z) / y)
	else:
		tmp = ((x + 4.0) - (x * z)) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.52)
		tmp = Float64(x * Float64(Float64(-1.0 + z) / y));
	else
		tmp = Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.52)
		tmp = x * ((-1.0 + z) / y);
	else
		tmp = ((x + 4.0) - (x * z)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.52], N[(x * N[(N[(-1.0 + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.52000000000000002

   1. Initial program 77.7%

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

      1. fabs-sub 77.7%

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

      2. associate-*l/ 81.1%

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

      3. associate-*r/ 85.6%

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

      4. fmm-def 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. +-commutative 93.5%

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

      7. distribute-neg-in 93.5%

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

      8. unsub-neg 93.5%

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

      9. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

      1. add-sqr-sqrt 52.1%

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

      2. fabs-sqr 52.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. fma-undefine 44.8%

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

      5. associate-*r/ 41.9%

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

      6. associate-*l/ 38.5%

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

      7. div-inv 38.5%

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

      8. sub-neg 38.5%

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

      9. metadata-eval 38.5%

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

      10. distribute-neg-in 38.5%

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

      11. +-commutative 38.5%

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

      12. cancel-sign-sub-inv 38.5%

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

      13. div-inv 38.5%

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

      14. associate-*l/ 41.9%

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

      15. sub-div 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/l* 52.7%

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

      2. sub-neg 52.7%

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

      3. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   if -0.52000000000000002 < x

   1. Initial program 90.2%

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

      1. fabs-sub 90.2%

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

      2. associate-*l/ 93.9%

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

      3. associate-*r/ 92.9%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. fma-undefine 92.9%

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

      2. associate-*r/ 93.9%

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

      3. associate-*l/ 90.2%

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

      4. div-inv 90.1%

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

      5. sub-neg 90.1%

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

      6. metadata-eval 90.1%

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

      7. distribute-neg-in 90.1%

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

      8. +-commutative 90.1%

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

      9. cancel-sign-sub-inv 90.1%

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

      10. div-inv 90.2%

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

      11. fabs-sub 90.2%

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

      12. add-sqr-sqrt 46.7%

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

      13. fabs-sqr 46.7%

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

      14. add-sqr-sqrt 47.7%

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

      15. associate-*l/ 50.5%

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

      16. sub-div 52.0%

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

   6. Applied egg-rr 52.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 4\right) - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -10.2) (/ (- x) y) (if (<= x 4.0) (/ 4.0 y) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -10.2) {
		tmp = -x / y;
	} else if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-10.2d0)) then
        tmp = -x / y
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -10.2) {
		tmp = -x / y;
	} else if (x <= 4.0) {
		tmp = 4.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -10.2:
		tmp = -x / y
	elif x <= 4.0:
		tmp = 4.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -10.2)
		tmp = Float64(Float64(-x) / y);
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -10.2)
		tmp = -x / y;
	elseif (x <= 4.0)
		tmp = 4.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -10.2], N[((-x) / y), $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.199999999999999

   1. Initial program 77.4%

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

      1. fabs-sub 77.4%

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

      2. associate-*l/ 80.8%

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

      3. associate-*r/ 85.4%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. add-sqr-sqrt 51.3%

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

      2. fabs-sqr 51.3%

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

      3. add-sqr-sqrt 52.0%

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

      4. fma-undefine 43.9%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 37.5%

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

      7. div-inv 37.5%

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

      8. sub-neg 37.5%

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

      9. metadata-eval 37.5%

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

      10. distribute-neg-in 37.5%

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

      11. +-commutative 37.5%

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

      12. cancel-sign-sub-inv 37.5%

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

      13. div-inv 37.5%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 49.0%

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

   6. Applied egg-rr 49.0%

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

   7. Taylor expanded in z around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. distribute-lft-in 38.3%

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

      3. metadata-eval 38.3%

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

      4. neg-mul-1 38.3%

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

      5. sub-neg 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in x around inf 38.3%

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

       1. neg-mul-1 38.3%

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

   12. Simplified 38.3%

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

   if -10.199999999999999 < x < 4

   1. Initial program 96.0%

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

   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 54.2%

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

      2. fabs-sqr 54.2%

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

      3. add-sqr-sqrt 55.5%

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

      4. associate-*l/ 55.5%

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

      5. clear-num 55.5%

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

      6. mul-1-neg 55.5%

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

   6. Applied egg-rr 55.5%

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

   7. Taylor expanded in x around 0 40.8%

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

   if 4 < x

   1. Initial program 80.8%

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

   3. Simplified 92.1%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.4%

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

      4. associate-*l/ 45.5%

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

      5. clear-num 45.5%

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

      6. mul-1-neg 45.5%

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

   6. Applied egg-rr 45.5%

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

   7. Taylor expanded in z around 0 28.2%

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

      1. +-commutative 28.2%

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

   9. Simplified 28.2%

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

   10. Taylor expanded in x around inf 28.3%

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

Alternative 10: 39.5% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.52) (* x (/ (+ -1.0 z) y)) (/ (+ x 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.52) {
		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 <= (-0.52d0)) 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 <= -0.52) {
		tmp = x * ((-1.0 + z) / y);
	} else {
		tmp = (x + 4.0) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.52:
		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 <= -0.52)
		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 <= -0.52)
		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, -0.52], 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 < -0.52000000000000002

   1. Initial program 77.7%

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

      1. fabs-sub 77.7%

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

      2. associate-*l/ 81.1%

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

      3. associate-*r/ 85.6%

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

      4. fmm-def 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. +-commutative 93.5%

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

      7. distribute-neg-in 93.5%

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

      8. unsub-neg 93.5%

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

      9. metadata-eval 93.5%

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

   3. Simplified 93.5%

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

      1. add-sqr-sqrt 52.1%

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

      2. fabs-sqr 52.1%

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

      3. add-sqr-sqrt 52.8%

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

      4. fma-undefine 44.8%

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

      5. associate-*r/ 41.9%

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

      6. associate-*l/ 38.5%

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

      7. div-inv 38.5%

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

      8. sub-neg 38.5%

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

      9. metadata-eval 38.5%

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

      10. distribute-neg-in 38.5%

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

      11. +-commutative 38.5%

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

      12. cancel-sign-sub-inv 38.5%

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

      13. div-inv 38.5%

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

      14. associate-*l/ 41.9%

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

      15. sub-div 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/l* 52.7%

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

      2. sub-neg 52.7%

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

      3. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   if -0.52000000000000002 < x

   1. Initial program 90.2%

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

   3. Simplified 96.9%

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

      1. add-sqr-sqrt 51.0%

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

      2. fabs-sqr 51.0%

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

      3. add-sqr-sqrt 52.0%

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

      4. associate-*l/ 52.0%

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

      5. clear-num 52.0%

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

      6. mul-1-neg 52.0%

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

   6. Applied egg-rr 52.0%

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

   7. Taylor expanded in z around 0 36.3%

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

      1. +-commutative 36.3%

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

   9. Simplified 36.3%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;x \cdot \frac{-1 + z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.1% accurate, 11.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 77.4%

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

      1. fabs-sub 77.4%

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

      2. associate-*l/ 80.8%

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

      3. associate-*r/ 85.4%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. add-sqr-sqrt 51.3%

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

      2. fabs-sqr 51.3%

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

      3. add-sqr-sqrt 52.0%

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

      4. fma-undefine 43.9%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 37.5%

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

      7. div-inv 37.5%

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

      8. sub-neg 37.5%

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

      9. metadata-eval 37.5%

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

      10. distribute-neg-in 37.5%

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

      11. +-commutative 37.5%

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

      12. cancel-sign-sub-inv 37.5%

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

      13. div-inv 37.5%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 49.0%

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

   6. Applied egg-rr 49.0%

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

   7. Taylor expanded in z around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. distribute-lft-in 38.3%

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

      3. metadata-eval 38.3%

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

      4. neg-mul-1 38.3%

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

      5. sub-neg 38.3%

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

   9. Simplified 38.3%

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

   if -4 < x

   1. Initial program 90.3%

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

   3. Simplified 96.9%

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

      1. add-sqr-sqrt 50.8%

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

      2. fabs-sqr 50.8%

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

      3. add-sqr-sqrt 51.7%

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

      4. associate-*l/ 51.7%

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

      5. clear-num 51.7%

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

      6. mul-1-neg 51.7%

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

   6. Applied egg-rr 51.7%

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

   7. Taylor expanded in z around 0 36.2%

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

      1. +-commutative 36.2%

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

   9. Simplified 36.2%

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

Alternative 12: 35.0% accurate, 11.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 77.4%

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

      1. fabs-sub 77.4%

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

      2. associate-*l/ 80.8%

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

      3. associate-*r/ 85.4%

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

      4. fmm-def 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

      1. add-sqr-sqrt 51.3%

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

      2. fabs-sqr 51.3%

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

      3. add-sqr-sqrt 52.0%

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

      4. fma-undefine 43.9%

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

      5. associate-*r/ 40.9%

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

      6. associate-*l/ 37.5%

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

      7. div-inv 37.5%

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

      8. sub-neg 37.5%

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

      9. metadata-eval 37.5%

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

      10. distribute-neg-in 37.5%

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

      11. +-commutative 37.5%

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

      12. cancel-sign-sub-inv 37.5%

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

      13. div-inv 37.5%

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

      14. associate-*l/ 40.9%

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

      15. sub-div 49.0%

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

   6. Applied egg-rr 49.0%

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

   7. Taylor expanded in z around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. distribute-lft-in 38.3%

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

      3. metadata-eval 38.3%

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

      4. neg-mul-1 38.3%

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

      5. sub-neg 38.3%

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

   9. Simplified 38.3%

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

   10. Taylor expanded in x around inf 38.3%

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

       1. neg-mul-1 38.3%

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

   12. Simplified 38.3%

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

   if -4 < x

   1. Initial program 90.3%

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

   3. Simplified 96.9%

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

      1. add-sqr-sqrt 50.8%

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

      2. fabs-sqr 50.8%

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

      3. add-sqr-sqrt 51.7%

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

      4. associate-*l/ 51.7%

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

      5. clear-num 51.7%

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

      6. mul-1-neg 51.7%

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

   6. Applied egg-rr 51.7%

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

   7. Taylor expanded in z around 0 36.2%

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

      1. +-commutative 36.2%

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

   9. Simplified 36.2%

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

Alternative 13: 27.5% 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 89.7%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 51.6%

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

      2. fabs-sqr 51.6%

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

      3. add-sqr-sqrt 52.6%

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

      4. associate-*l/ 52.7%

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

      5. clear-num 52.6%

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

      6. mul-1-neg 52.6%

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

   6. Applied egg-rr 52.6%

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

   7. Taylor expanded in x around 0 28.6%

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

   if 4 < x

   1. Initial program 80.8%

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

   3. Simplified 92.1%

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

      1. add-sqr-sqrt 45.2%

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

      2. fabs-sqr 45.2%

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

      3. add-sqr-sqrt 45.4%

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

      4. associate-*l/ 45.5%

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

      5. clear-num 45.5%

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

      6. mul-1-neg 45.5%

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

   6. Applied egg-rr 45.5%

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

   7. Taylor expanded in z around 0 28.2%

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

      1. +-commutative 28.2%

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

   9. Simplified 28.2%

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

   10. Taylor expanded in x around inf 28.3%

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

Alternative 14: 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 87.1%

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

3. Simplified 96.5%

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

   1. add-sqr-sqrt 49.7%

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

   2. fabs-sqr 49.7%

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

   3. add-sqr-sqrt 50.5%

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

   4. associate-*l/ 50.6%

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

   5. clear-num 50.6%

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

   6. mul-1-neg 50.6%

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

6. Applied egg-rr 50.6%

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

7. Taylor expanded in x around 0 21.3%

   \[\leadsto \color{blue}{\frac{4}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024166 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))