fabs fraction 1

Percentage Accurate: 92.1% → 99.9%

Time: 7.0s

Alternatives: 9

Speedup: 1.2×

• 20.5% 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: 92.1% 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: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 500000000:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-x, \frac{z}{y\_m}, \frac{4 + x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 500000000.0)
   (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 <= 500000000.0) {
		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 <= 500000000.0)
		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, 500000000.0], 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 < 5e8

   1. Initial program 89.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 98.5%

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

   if 5e8 < y

   1. Initial program 99.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 2: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) (- 1.0 z)))))
   (if (<= x -1.55)
     t_0
     (if (<= x 0.07) (fabs (/ (fma (- z) x 4.0) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 0.07) {
		tmp = fabs((fma(-z, x, 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)))
	tmp = 0.0
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 0.07)
		tmp = abs(Float64(fma(Float64(-z), x, 4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.55], t$95$0, If[LessEqual[x, 0.07], N[Abs[N[(N[((-z) * x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.070000000000000007 < x

   1. Initial program 84.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.55000000000000004 < x < 0.070000000000000007

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(-z, x, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-69}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) (- 1.0 z)))))
   (if (<= x -2.2e-5) t_0 (if (<= x 9e-69) (fabs (/ (+ 4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (x <= -2.2e-5) {
		tmp = t_0;
	} else if (x <= 9e-69) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs(((x / y_m) * (1.0d0 - z)))
    if (x <= (-2.2d-5)) then
        tmp = t_0
    else if (x <= 9d-69) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (x <= -2.2e-5) {
		tmp = t_0;
	} else if (x <= 9e-69) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x / y_m) * (1.0 - z)))
	tmp = 0
	if x <= -2.2e-5:
		tmp = t_0
	elif x <= 9e-69:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)))
	tmp = 0.0
	if (x <= -2.2e-5)
		tmp = t_0;
	elseif (x <= 9e-69)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x / y_m) * (1.0 - z)));
	tmp = 0.0;
	if (x <= -2.2e-5)
		tmp = t_0;
	elseif (x <= 9e-69)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.2e-5], t$95$0, If[LessEqual[x, 9e-69], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-5 or 9.00000000000000019e-69 < x

   1. Initial program 86.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -2.1999999999999999e-5 < x < 9.00000000000000019e-69

   1. Initial program 97.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-69}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{-x}{y\_m} \cdot z\right|\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+62}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ (- x) y_m) z))))
   (if (<= z -2.15e+98)
     t_0
     (if (<= z 1.18e+62) (fabs (/ (+ 4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((-x / y_m) * z));
	double tmp;
	if (z <= -2.15e+98) {
		tmp = t_0;
	} else if (z <= 1.18e+62) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs(((-x / y_m) * z))
    if (z <= (-2.15d+98)) then
        tmp = t_0
    else if (z <= 1.18d+62) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((-x / y_m) * z));
	double tmp;
	if (z <= -2.15e+98) {
		tmp = t_0;
	} else if (z <= 1.18e+62) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((-x / y_m) * z))
	tmp = 0
	if z <= -2.15e+98:
		tmp = t_0
	elif z <= 1.18e+62:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(Float64(-x) / y_m) * z))
	tmp = 0.0
	if (z <= -2.15e+98)
		tmp = t_0;
	elseif (z <= 1.18e+62)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((-x / y_m) * z));
	tmp = 0.0;
	if (z <= -2.15e+98)
		tmp = t_0;
	elseif (z <= 1.18e+62)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[((-x) / y$95$m), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2.15e+98], t$95$0, If[LessEqual[z, 1.18e+62], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1500000000000001e98 or 1.18000000000000001e62 < z

   1. Initial program 88.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   if -2.1500000000000001e98 < z < 1.18000000000000001e62

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{-x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+62}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-x}{y} \cdot z\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+62}:\\ \;\;\;\;\left|\frac{4 + x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ z y_m) x))))
   (if (<= z -2.15e+98)
     t_0
     (if (<= z 1.18e+62) (fabs (/ (+ 4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((z / y_m) * x));
	double tmp;
	if (z <= -2.15e+98) {
		tmp = t_0;
	} else if (z <= 1.18e+62) {
		tmp = fabs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs(((z / y_m) * x))
    if (z <= (-2.15d+98)) then
        tmp = t_0
    else if (z <= 1.18d+62) then
        tmp = abs(((4.0d0 + x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((z / y_m) * x));
	double tmp;
	if (z <= -2.15e+98) {
		tmp = t_0;
	} else if (z <= 1.18e+62) {
		tmp = Math.abs(((4.0 + x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((z / y_m) * x))
	tmp = 0
	if z <= -2.15e+98:
		tmp = t_0
	elif z <= 1.18e+62:
		tmp = math.fabs(((4.0 + x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(z / y_m) * x))
	tmp = 0.0
	if (z <= -2.15e+98)
		tmp = t_0;
	elseif (z <= 1.18e+62)
		tmp = abs(Float64(Float64(4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((z / y_m) * x));
	tmp = 0.0;
	if (z <= -2.15e+98)
		tmp = t_0;
	elseif (z <= 1.18e+62)
		tmp = abs(((4.0 + x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2.15e+98], t$95$0, If[LessEqual[z, 1.18e+62], N[Abs[N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1500000000000001e98 or 1.18000000000000001e62 < z

   1. Initial program 88.4%

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

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 74.8

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

   7. Applied rewrites 74.8%

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

   if -2.1500000000000001e98 < z < 1.18000000000000001e62

   1. Initial program 92.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.9%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -7e+26)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (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 <= -7e+26) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} 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 <= -7e+26)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	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[LessEqual[x, -7e+26], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $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 < -6.9999999999999998e26

   1. Initial program 84.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -6.9999999999999998e26 < x

   1. Initial program 93.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. distribute-rgt-out N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(1 - z, x, 4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -10.5) t_0 (if (<= x 4.0) (fabs (/ 4.0 y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-10.5d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -10.5) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -10.5:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -10.5)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -10.5], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 83.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. mul-1-neg N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.9%

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

   if -10.5 < x < 4

   1. Initial program 97.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

Alternative 8: 69.6% 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 91.2%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. associate-*l* N/A

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

   11. *-rgt-identity N/A

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

   12. associate-*r/ N/A

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

   13. distribute-rgt-out N/A

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

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

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

   15. *-commutative N/A

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

   16. associate-/l* N/A

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 70.2%

   \[\leadsto \left|\frac{4 + x}{y}\right| \]
9. Add Preprocessing

Alternative 9: 33.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{x}{y\_m}\right| \end{array} \]

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs((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((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((x / y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return math.fabs((x / y_m))

Julia

y_m = abs(y)
function code(x, y_m, z)
	return abs(Float64(x / y_m))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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. *-commutative N/A

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

   6. associate-/l* N/A

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

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

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

   8. mul-1-neg N/A

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

   9. distribute-rgt1-in N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 59.8

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

5. Applied rewrites 59.8%

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

7. Applied rewrites 59.8%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 31.5%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))