fabs fraction 1

Percentage Accurate: 91.8% → 99.5%

Time: 8.0s

Alternatives: 15

Speedup: 1.1×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5.2e-85)
   (fabs (/ (fma (- 1.0 z) x 4.0) y_m))
   (fabs (fma (- x) (/ z y_m) (/ (+ 4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5.2e-85) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y_m));
	} else {
		tmp = fabs(fma(-x, (z / y_m), ((4.0 + x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5.2e-85)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y_m));
	else
		tmp = abs(fma(Float64(-x), Float64(z / y_m), Float64(Float64(4.0 + x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5.2e-85], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision] + N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000023e-85

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 95.0%

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

   if 5.20000000000000023e-85 < y

   1. Initial program 97.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 5e-308)
     (fabs (* (- 1.0 z) (/ x y_m)))
     (if (<= t_0 INFINITY) (/ (fma x (- 1.0 z) 4.0) y_m) (fabs (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (1.0 - z), 4.0) / y_m;
	} else {
		tmp = fabs((x / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= 5e-308)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(1.0 - z), 4.0) / y_m);
	else
		tmp = abs(Float64(x / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-308], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.99999999999999955e-308

   1. Initial program 96.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 57.8

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

   5. Applied rewrites 57.8%

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

   if 4.99999999999999955e-308 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 98.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 98.3

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

   9. Applied rewrites 93.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

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

Alternative 3: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\left|\frac{-1 + z}{y\_m} \cdot x\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 2e-280)
     (fabs (* (/ (+ -1.0 z) y_m) x))
     (if (<= t_0 INFINITY) (/ (fma x (- 1.0 z) 4.0) y_m) (fabs (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= 2e-280) {
		tmp = fabs((((-1.0 + z) / y_m) * x));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (1.0 - z), 4.0) / y_m;
	} else {
		tmp = fabs((x / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= 2e-280)
		tmp = abs(Float64(Float64(Float64(-1.0 + z) / y_m) * x));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(1.0 - z), 4.0) / y_m);
	else
		tmp = abs(Float64(x / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-280], N[Abs[N[(N[(N[(-1.0 + z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 1.9999999999999999e-280

   1. Initial program 95.5%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-add-rev N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. lower-+.f64 60.3

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

   7. Applied rewrites 60.3%

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

   if 1.9999999999999999e-280 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 94.1%

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

   7. Applied rewrites 99.1%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 99.1

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

   9. Applied rewrites 93.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

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

Alternative 4: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 5e-308)
     (fabs (/ (- -4.0 x) y_m))
     (if (<= t_0 INFINITY) (/ (fma x (- 1.0 z) 4.0) y_m) (fabs (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (1.0 - z), 4.0) / y_m;
	} else {
		tmp = fabs((x / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= 5e-308)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(1.0 - z), 4.0) / y_m);
	else
		tmp = abs(Float64(x / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-308], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.99999999999999955e-308

   1. Initial program 96.2%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 4.99999999999999955e-308 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < +inf.0

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 98.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 98.3

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

   9. Applied rewrites 93.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

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

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x}{y\_m} \cdot z - \frac{x + 4}{y\_m}\right| \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(1 - z, \frac{x}{y\_m}, \frac{4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= (fabs (- (* (/ x y_m) z) (/ (+ x 4.0) y_m))) 5e-72)
   (fabs (/ (fma (- 1.0 z) x 4.0) y_m))
   (fabs (fma (- 1.0 z) (/ x y_m) (/ 4.0 y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (fabs((((x / y_m) * z) - ((x + 4.0) / y_m))) <= 5e-72) {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y_m));
	} else {
		tmp = fabs(fma((1.0 - z), (x / y_m), (4.0 / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (abs(Float64(Float64(Float64(x / y_m) * z) - Float64(Float64(x + 4.0) / y_m))) <= 5e-72)
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y_m));
	else
		tmp = abs(fma(Float64(1.0 - z), Float64(x / y_m), Float64(4.0 / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[N[Abs[N[(N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-72], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision] + N[(4.0 / y$95$m), $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))) < 4.9999999999999996e-72

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.8%

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

   if 4.9999999999999996e-72 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 94.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right| \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(1 - z, \frac{x}{y}, \frac{4}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -82000000000000 \lor \neg \left(x \leq 5 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -82000000000000.0) (not (<= x 5e+16)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (fma (- 1.0 z) x 4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -82000000000000.0) || !(x <= 5e+16)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs((fma((1.0 - z), x, 4.0) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -82000000000000.0) || !(x <= 5e+16))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(fma(Float64(1.0 - z), x, 4.0) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -82000000000000.0], N[Not[LessEqual[x, 5e+16]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2e13 or 5e16 < x

   1. Initial program 89.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -8.2e13 < x < 5e16

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate--l+ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -82000000000000 \lor \neg \left(x \leq 5 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -11 \lor \neg \left(x \leq 0.0155\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -11.0) (not (<= x 0.0155)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (- (* z x) 4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -11.0) || !(x <= 0.0155)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs((((z * x) - 4.0) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-11.0d0)) .or. (.not. (x <= 0.0155d0))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs((((z * x) - 4.0d0) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -11.0) || !(x <= 0.0155)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs((((z * x) - 4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -11.0) or not (x <= 0.0155):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs((((z * x) - 4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -11.0) || !(x <= 0.0155))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(Float64(z * x) - 4.0) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -11.0) || ~((x <= 0.0155)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs((((z * x) - 4.0) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -11.0], N[Not[LessEqual[x, 0.0155]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(z * x), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -11 or 0.0155 < x

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -11 < x < 0.0155

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 98.5

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

   7. Applied rewrites 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11 \lor \neg \left(x \leq 0.0155\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+27} \lor \neg \left(z \leq 1.08 \cdot 10^{+71}\right):\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.55e+27) (not (<= z 1.08e+71)))
   (fabs (* (- x) (/ z y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.55e+27) || !(z <= 1.08e+71)) {
		tmp = fabs((-x * (z / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.55d+27)) .or. (.not. (z <= 1.08d+71))) then
        tmp = abs((-x * (z / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.55e+27) || !(z <= 1.08e+71)) {
		tmp = Math.abs((-x * (z / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.55e+27) or not (z <= 1.08e+71):
		tmp = math.fabs((-x * (z / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.55e+27) || !(z <= 1.08e+71))
		tmp = abs(Float64(Float64(-x) * Float64(z / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.55e+27) || ~((z <= 1.08e+71)))
		tmp = abs((-x * (z / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.55e+27], N[Not[LessEqual[z, 1.08e+71]], $MachinePrecision]], N[Abs[N[((-x) * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999998e27 or 1.08e71 < z

   1. Initial program 92.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 78.1%

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

   if -1.54999999999999998e27 < z < 1.08e71

   1. Initial program 92.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+27} \lor \neg \left(z \leq 1.08 \cdot 10^{+71}\right):\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+27} \lor \neg \left(z \leq 9.6 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, x, 4\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -4.3e+27) (not (<= z 9.6e+35)))
   (/ (fma (- z) x 4.0) y_m)
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -4.3e+27) || !(z <= 9.6e+35)) {
		tmp = fma(-z, x, 4.0) / y_m;
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -4.3e+27) || !(z <= 9.6e+35))
		tmp = Float64(fma(Float64(-z), x, 4.0) / y_m);
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -4.3e+27], N[Not[LessEqual[z, 9.6e+35]], $MachinePrecision]], N[(N[((-z) * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.30000000000000008e27 or 9.60000000000000058e35 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 88.5

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

   7. Applied rewrites 88.5%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 42.3

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 42.3

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

   9. Applied rewrites 42.3%

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

   if -4.30000000000000008e27 < z < 9.60000000000000058e35

   1. Initial program 93.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 97.9

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

   7. Applied rewrites 97.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+27} \lor \neg \left(z \leq 9.6 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+63} \lor \neg \left(z \leq 8 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{-z}{y\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -1.3e+63) (not (<= z 8e+136)))
   (* (/ (- z) y_m) x)
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.3e+63) || !(z <= 8e+136)) {
		tmp = (-z / y_m) * x;
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.3d+63)) .or. (.not. (z <= 8d+136))) then
        tmp = (-z / y_m) * x
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -1.3e+63) || !(z <= 8e+136)) {
		tmp = (-z / y_m) * x;
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -1.3e+63) or not (z <= 8e+136):
		tmp = (-z / y_m) * x
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -1.3e+63) || !(z <= 8e+136))
		tmp = Float64(Float64(Float64(-z) / y_m) * x);
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((z <= -1.3e+63) || ~((z <= 8e+136)))
		tmp = (-z / y_m) * x;
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -1.3e+63], N[Not[LessEqual[z, 8e+136]], $MachinePrecision]], N[(N[((-z) / y$95$m), $MachinePrecision] * x), $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3000000000000001e63 or 8.00000000000000047e136 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 37.3

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

   7. Applied rewrites 37.3%

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

   9. Applied rewrites 39.5%

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

   if -1.3000000000000001e63 < z < 8.00000000000000047e136

   1. Initial program 93.4%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+63} \lor \neg \left(z \leq 8 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{-z}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 7.8e-11) (fabs (/ (- -4.0 x) y_m)) (* (/ x y_m) (- 1.0 z))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 7.8e-11) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = (x / y_m) * (1.0 - z);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 7.8d-11) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = (x / y_m) * (1.0d0 - z)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 7.8e-11) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = (x / y_m) * (1.0 - z);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 7.8e-11:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = (x / y_m) * (1.0 - z)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 7.8e-11)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = Float64(Float64(x / y_m) * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 7.8e-11)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = (x / y_m) * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 7.8e-11], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.80000000000000021e-11

   1. Initial program 93.4%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 72.7

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

   7. Applied rewrites 72.7%

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

   if 7.80000000000000021e-11 < x

   1. Initial program 90.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 48.1

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

   7. Applied rewrites 48.1%

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

Alternative 12: 69.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -10.5) (fabs (/ x y_m)) (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -10.5) {
		tmp = fabs((x / y_m));
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-10.5d0)) then
        tmp = abs((x / y_m))
    else if (x <= 4.0d0) then
        tmp = 4.0d0 / y_m
    else
        tmp = x / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -10.5) {
		tmp = Math.abs((x / y_m));
	} else if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -10.5:
		tmp = math.fabs((x / y_m))
	elif x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -10.5)
		tmp = abs(Float64(x / y_m));
	elseif (x <= 4.0)
		tmp = Float64(4.0 / y_m);
	else
		tmp = Float64(x / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -10.5)
		tmp = abs((x / y_m));
	elseif (x <= 4.0)
		tmp = 4.0 / y_m;
	else
		tmp = x / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -10.5], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -10.5

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 69.6%

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

   if -10.5 < x < 4

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 33.2

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

   7. Applied rewrites 33.2%

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

   if 4 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 53.1%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 24.3

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

   13. Applied rewrites 24.3%

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

Alternative 13: 55.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{4}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (if (<= x 4.0) (/ 4.0 y_m) (/ x y_m)))

C

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

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 4.0) {
		tmp = 4.0 / y_m;
	} else {
		tmp = x / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 4.0:
		tmp = 4.0 / y_m
	else:
		tmp = x / y_m
	return tmp

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 4.0], N[(4.0 / y$95$m), $MachinePrecision], N[(x / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 22.9

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

   7. Applied rewrites 22.9%

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

   if 4 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. lower--.f64 N/A

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

      19. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 53.1%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. sqrt-prod N/A

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

       4. rem-square-sqrt 24.3

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

   13. Applied rewrites 24.3%

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

Alternative 14: 70.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{-4 - x}{y\_m}\right| \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (- -4.0 x) y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.6%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-prod N/A

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

   3. rem-sqrt-square-rev N/A

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

   4. lift--.f64 N/A

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

   5. fabs-sub N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. sqrt-prod N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 47.5%

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

5. Taylor expanded in z around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower--.f64 67.6

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

7. Applied rewrites 67.6%

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

Alternative 15: 18.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{x}{y\_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (/ x y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x / y_m;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x / y_m
end function

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return x / y_m

Julia

y_m = abs(y)
function code(x, y_m, z)
	return Float64(x / y_m)
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x / y$95$m), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. div-sub N/A

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

   6. fp-cancel-sub-sign-inv N/A

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

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

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt1-in N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. fp-cancel-sign-sub-inv N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. lower--.f64 N/A

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

   19. lower-/.f64 61.9

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

5. Applied rewrites 61.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 32.9%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 32.9%

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

    1. lift-fabs.f64 N/A

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

    2. rem-sqrt-square-rev N/A

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

    3. sqrt-prod N/A

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

    4. rem-square-sqrt 18.0

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

13. Applied rewrites 18.0%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))