fabs fraction 1

Percentage Accurate: 92.4% → 99.8%

Time: 4.6s

Alternatives: 10

Speedup: 1.0×

• 79.8% of points produce a very large (infinite) output. You may want to add a precondition.

• Could not converge on a ground truth

  1. x = -1.2239212893689534e-70
  2. y = 1.20626951932792
  3. z = 3.849098840970663e+268

Specification

Math

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.4% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{\left|y\right|}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{\left|y\right|} - x \cdot \left(\frac{1}{\left|y\right|} \cdot z\right)\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= (fabs y) 5e+58)
  (fabs (/ (fma z x (- -4.0 x)) (fabs y)))
  (fabs (- (/ (+ x 4.0) (fabs y)) (* x (* (/ 1.0 (fabs y)) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (fabs(y) <= 5e+58) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / fabs(y)));
	} else {
		tmp = fabs((((x + 4.0) / fabs(y)) - (x * ((1.0 / fabs(y)) * z))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (abs(y) <= 5e+58)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / abs(y)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / abs(y)) - Float64(x * Float64(Float64(1.0 / abs(y)) * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Abs[y], $MachinePrecision], 5e+58], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[(1.0 / N[Abs[y], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e58

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

   if 4.9999999999999999e58 < y

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (* (/ (- 1.0 z) y) x))))
  (if (<= x -1.3e+189)
    t_0
    (if (<= x 5e+46) (fabs (/ (fma z x (- -4.0 x)) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((1.0 - z) / y) * x));
	double tmp;
	if (x <= -1.3e+189) {
		tmp = t_0;
	} else if (x <= 5e+46) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(1.0 - z) / y) * x))
	tmp = 0.0
	if (x <= -1.3e+189)
		tmp = t_0;
	elseif (x <= 5e+46)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.3e+189], t$95$0, If[LessEqual[x, 5e+46], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2999999999999999e189 or 5.0000000000000002e46 < x

   1. Initial program 92.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.1%

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.1%

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 61.1%

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

   6. Applied rewrites 61.1%

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

   if -1.2999999999999999e189 < x < 5.0000000000000002e46

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{if}\;x \leq -100000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 86000000:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (* (/ (- 1.0 z) y) x))))
  (if (<= x -100000.0)
    t_0
    (if (<= x 86000000.0) (fabs (/ (fma z x -4.0) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((1.0 - z) / y) * x));
	double tmp;
	if (x <= -100000.0) {
		tmp = t_0;
	} else if (x <= 86000000.0) {
		tmp = fabs((fma(z, x, -4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(1.0 - z) / y) * x))
	tmp = 0.0
	if (x <= -100000.0)
		tmp = t_0;
	elseif (x <= 86000000.0)
		tmp = abs(Float64(fma(z, x, -4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -100000.0], t$95$0, If[LessEqual[x, 86000000.0], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e5 or 8.6e7 < x

   1. Initial program 92.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.1%

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.1%

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 61.1%

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

   6. Applied rewrites 61.1%

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

   if -1e5 < x < 8.6e7

   1. Initial program 92.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-neg.f64 95.9%

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 73.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. fabs-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-/.f64 73.7%

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

   8. Applied rewrites 73.7%

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

Alternative 4: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{x + 4}{y} - \frac{x}{y} \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (- (/ (+ x 4.0) y) (* (/ x y) z))))
  (if (<= t_0 -2e-88) (fabs t_0) (fabs (/ (fma z x (- -4.0 x)) y)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - ((x / y) * z);
	double tmp;
	if (t_0 <= -2e-88) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z))
	tmp = 0.0
	if (t_0 <= -2e-88)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-88], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1.9999999999999999e-88

   1. Initial program 92.4%

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

   if -1.9999999999999999e-88 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

Alternative 5: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{if}\;z \leq -600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.2:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ (fma z x -4.0) y))))
  (if (<= z -600000000.0)
    t_0
    (if (<= z 5.2) (fabs (/ (- -4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((fma(z, x, -4.0) / y));
	double tmp;
	if (z <= -600000000.0) {
		tmp = t_0;
	} else if (z <= 5.2) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = abs(Float64(fma(z, x, -4.0) / y))
	tmp = 0.0
	if (z <= -600000000.0)
		tmp = t_0;
	elseif (z <= 5.2)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -600000000.0], t$95$0, If[LessEqual[z, 5.2], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e8 or 5.2000000000000002 < z

   1. Initial program 92.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-neg.f64 95.9%

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 73.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. fabs-neg N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-/.f64 73.7%

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

   8. Applied rewrites 73.7%

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

   if -6e8 < z < 5.2000000000000002

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0%

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

   8. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\frac{-4 + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

      6. sub-flip N/A

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

      7. lower--.f64 70.0%

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

   10. Applied rewrites 70.0%

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

Alternative 6: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{-x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-z}{y} \cdot x\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= z -3.4e+52)
  (fabs (* (/ (- x) y) z))
  (if (<= z 4.4e+49)
    (fabs (/ (- -4.0 x) y))
    (fabs (* (/ (- z) y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+52) {
		tmp = fabs(((-x / y) * z));
	} else if (z <= 4.4e+49) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((-z / y) * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.4d+52)) then
        tmp = abs(((-x / y) * z))
    else if (z <= 4.4d+49) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((-z / y) * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+52) {
		tmp = Math.abs(((-x / y) * z));
	} else if (z <= 4.4e+49) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((-z / y) * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.4e+52:
		tmp = math.fabs(((-x / y) * z))
	elif z <= 4.4e+49:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((-z / y) * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e+52)
		tmp = abs(Float64(Float64(Float64(-x) / y) * z));
	elseif (z <= 4.4e+49)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(Float64(-z) / y) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.4e+52)
		tmp = abs(((-x / y) * z));
	elseif (z <= 4.4e+49)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((-z / y) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e+52], N[Abs[N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.4e+49], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[((-z) / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4e52

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.4%

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

   4. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right| \]

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-neg-frac N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(x\right)}{y} \cdot z\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(x\right)}{y} \cdot z\right| \]

      10. lower-neg.f64 43.1%

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

   6. Applied rewrites 43.1%

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

   if -3.4e52 < z < 4.4000000000000001e49

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0%

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

   8. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\frac{-4 + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

      6. sub-flip N/A

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

      7. lower--.f64 70.0%

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

   10. Applied rewrites 70.0%

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

   if 4.4000000000000001e49 < z

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.4%

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

   4. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right| \]

      6. distribute-lft-neg-out N/A

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

      7. mult-flip N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(x \cdot \frac{1}{y}\right)\right) \cdot z\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\left(\mathsf{neg}\left(x \cdot \frac{1}{y}\right)\right) \cdot z\right| \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \left|\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{y}\right) \cdot z\right| \]

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. distribute-rgt-neg-in N/A

          \[\leadsto \left|\mathsf{neg}\left(\frac{z}{y} \cdot x\right)\right| \]

      19. distribute-lft-neg-in N/A

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

      20. lower-*.f64 N/A

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

      21. lift-/.f64 N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot x\right| \]

      22. distribute-neg-frac N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(z\right)}{y} \cdot x\right| \]

      23. lower-/.f64 N/A

          \[\leadsto \left|\frac{\mathsf{neg}\left(z\right)}{y} \cdot x\right| \]

      24. lower-neg.f64 39.3%

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

   6. Applied rewrites 39.3%

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

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{-x}{y} \cdot z\right|\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (* (/ (- x) y) z))))
  (if (<= z -3.4e+52)
    t_0
    (if (<= z 4.4e+49) (fabs (/ (- -4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((-x / y) * z));
	double tmp;
	if (z <= -3.4e+52) {
		tmp = t_0;
	} else if (z <= 4.4e+49) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((-x / y) * z))
    if (z <= (-3.4d+52)) then
        tmp = t_0
    else if (z <= 4.4d+49) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((-x / y) * z));
	double tmp;
	if (z <= -3.4e+52) {
		tmp = t_0;
	} else if (z <= 4.4e+49) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((-x / y) * z))
	tmp = 0
	if z <= -3.4e+52:
		tmp = t_0
	elif z <= 4.4e+49:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(-x) / y) * z))
	tmp = 0.0
	if (z <= -3.4e+52)
		tmp = t_0;
	elseif (z <= 4.4e+49)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((-x / y) * z));
	tmp = 0.0;
	if (z <= -3.4e+52)
		tmp = t_0;
	elseif (z <= 4.4e+49)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -3.4e+52], t$95$0, If[LessEqual[z, 4.4e+49], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4e52 or 4.4000000000000001e49 < z

   1. Initial program 92.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 37.4%

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

   4. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\mathsf{neg}\left(\frac{x}{y} \cdot z\right)\right| \]

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-neg-frac N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(x\right)}{y} \cdot z\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(x\right)}{y} \cdot z\right| \]

      10. lower-neg.f64 43.1%

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

   6. Applied rewrites 43.1%

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

   if -3.4e52 < z < 4.4000000000000001e49

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0%

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

   8. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\frac{-4 + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

      6. sub-flip N/A

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

      7. lower--.f64 70.0%

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

   10. Applied rewrites 70.0%

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

Alternative 8: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x \cdot z}{y}\right|\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ (* x z) y))))
  (if (<= z -3.4e+52)
    t_0
    (if (<= z 1.65e+133) (fabs (/ (- -4.0 x) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x * z) / y));
	double tmp;
	if (z <= -3.4e+52) {
		tmp = t_0;
	} else if (z <= 1.65e+133) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * z) / y))
    if (z <= (-3.4d+52)) then
        tmp = t_0
    else if (z <= 1.65d+133) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((x * z) / y));
	double tmp;
	if (z <= -3.4e+52) {
		tmp = t_0;
	} else if (z <= 1.65e+133) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x * z) / y))
	tmp = 0
	if z <= -3.4e+52:
		tmp = t_0
	elif z <= 1.65e+133:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x * z) / y))
	tmp = 0.0
	if (z <= -3.4e+52)
		tmp = t_0;
	elseif (z <= 1.65e+133)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x * z) / y));
	tmp = 0.0;
	if (z <= -3.4e+52)
		tmp = t_0;
	elseif (z <= 1.65e+133)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -3.4e+52], t$95$0, If[LessEqual[z, 1.65e+133], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4e52 or 1.65e133 < z

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0%

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

   8. Applied rewrites 70.0%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 37.4%

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

   11. Applied rewrites 37.4%

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

   if -3.4e52 < z < 1.65e133

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0%

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

   3. Applied rewrites 92.0%

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0%

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

   8. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

         \[\leadsto \left|\frac{-4 + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

      6. sub-flip N/A

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

      7. lower--.f64 70.0%

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

   10. Applied rewrites 70.0%

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

Alternative 9: 70.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 92.0%

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

3. Applied rewrites 92.0%

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 70.0%

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

8. Applied rewrites 70.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. mul-1-neg N/A

      \[\leadsto \left|\frac{-4 + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

   6. sub-flip N/A

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

   7. lower--.f64 70.0%

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

10. Applied rewrites 70.0%

    \[\leadsto \left|\frac{-4 - \color{blue}{x}}{y}\right| \]
11. Add Preprocessing

Alternative 10: 40.6% accurate, 3.1× speedup

Math

\[\left|\frac{-4}{y}\right| \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 92.0%

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

3. Applied rewrites 92.0%

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 40.6%

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

8. Applied rewrites 40.6%

   \[\leadsto \left|\color{blue}{\frac{-4}{y}}\right| \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025326 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))