fabs fraction 1

Percentage Accurate: 91.6% → 98.2%

Time: 6.1s

Alternatives: 9

Speedup: 1.0×

• 21.2% 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: 91.6% 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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+40)
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+40) {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e+40)
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e+40], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000003e40

   1. Initial program 96.3%

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

      1. fabs-sub 96.3%

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

      2. associate-*l/ 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*l/ 99.9%

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

      5. *-commutative 99.9%

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

      6. fma-neg 99.9%

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

      7. distribute-neg-frac 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   if -1.00000000000000003e40 < y

   1. Initial program 90.5%

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

      1. associate-*l/ 94.1%

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

      2. sub-div 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 2: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ t_2 := \left|\frac{x \cdot z}{y}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+182}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y))) (t_1 (fabs (/ x y))) (t_2 (fabs (/ (* x z) y))))
   (if (<= x -1.65e+26)
     t_1
     (if (<= x -4.7e-23)
       t_2
       (if (<= x 2.2e-144)
         t_0
         (if (<= x 4e-111)
           t_2
           (if (<= x 1.5e-35)
             t_0
             (if (<= x 9.5e+182) (fabs (* z (/ x y))) t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((4.0 / y));
	double t_1 = fabs((x / y));
	double t_2 = fabs(((x * z) / y));
	double tmp;
	if (x <= -1.65e+26) {
		tmp = t_1;
	} else if (x <= -4.7e-23) {
		tmp = t_2;
	} else if (x <= 2.2e-144) {
		tmp = t_0;
	} else if (x <= 4e-111) {
		tmp = t_2;
	} else if (x <= 1.5e-35) {
		tmp = t_0;
	} else if (x <= 9.5e+182) {
		tmp = fabs((z * (x / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = abs((4.0d0 / y))
    t_1 = abs((x / y))
    t_2 = abs(((x * z) / y))
    if (x <= (-1.65d+26)) then
        tmp = t_1
    else if (x <= (-4.7d-23)) then
        tmp = t_2
    else if (x <= 2.2d-144) then
        tmp = t_0
    else if (x <= 4d-111) then
        tmp = t_2
    else if (x <= 1.5d-35) then
        tmp = t_0
    else if (x <= 9.5d+182) then
        tmp = abs((z * (x / y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((4.0 / y));
	double t_1 = Math.abs((x / y));
	double t_2 = Math.abs(((x * z) / y));
	double tmp;
	if (x <= -1.65e+26) {
		tmp = t_1;
	} else if (x <= -4.7e-23) {
		tmp = t_2;
	} else if (x <= 2.2e-144) {
		tmp = t_0;
	} else if (x <= 4e-111) {
		tmp = t_2;
	} else if (x <= 1.5e-35) {
		tmp = t_0;
	} else if (x <= 9.5e+182) {
		tmp = Math.abs((z * (x / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((4.0 / y))
	t_1 = math.fabs((x / y))
	t_2 = math.fabs(((x * z) / y))
	tmp = 0
	if x <= -1.65e+26:
		tmp = t_1
	elif x <= -4.7e-23:
		tmp = t_2
	elif x <= 2.2e-144:
		tmp = t_0
	elif x <= 4e-111:
		tmp = t_2
	elif x <= 1.5e-35:
		tmp = t_0
	elif x <= 9.5e+182:
		tmp = math.fabs((z * (x / y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(4.0 / y))
	t_1 = abs(Float64(x / y))
	t_2 = abs(Float64(Float64(x * z) / y))
	tmp = 0.0
	if (x <= -1.65e+26)
		tmp = t_1;
	elseif (x <= -4.7e-23)
		tmp = t_2;
	elseif (x <= 2.2e-144)
		tmp = t_0;
	elseif (x <= 4e-111)
		tmp = t_2;
	elseif (x <= 1.5e-35)
		tmp = t_0;
	elseif (x <= 9.5e+182)
		tmp = abs(Float64(z * Float64(x / y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((4.0 / y));
	t_1 = abs((x / y));
	t_2 = abs(((x * z) / y));
	tmp = 0.0;
	if (x <= -1.65e+26)
		tmp = t_1;
	elseif (x <= -4.7e-23)
		tmp = t_2;
	elseif (x <= 2.2e-144)
		tmp = t_0;
	elseif (x <= 4e-111)
		tmp = t_2;
	elseif (x <= 1.5e-35)
		tmp = t_0;
	elseif (x <= 9.5e+182)
		tmp = abs((z * (x / y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65e+26], t$95$1, If[LessEqual[x, -4.7e-23], t$95$2, If[LessEqual[x, 2.2e-144], t$95$0, If[LessEqual[x, 4e-111], t$95$2, If[LessEqual[x, 1.5e-35], t$95$0, If[LessEqual[x, 9.5e+182], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.64999999999999997e26 or 9.50000000000000002e182 < x

   1. Initial program 84.8%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-lft-in 89.3%

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

      5. associate-*r/ 89.5%

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

      6. *-rgt-identity 89.5%

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

      7. mul-1-neg 89.5%

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

      8. distribute-rgt-neg-in 89.5%

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

      9. unsub-neg 89.5%

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

      10. *-lft-identity 89.5%

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

      11. associate-/l* 89.3%

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

      12. *-commutative 89.3%

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

      13. associate-/r/ 84.6%

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

      14. div-sub 99.8%

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

      15. associate-/r/ 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around 0 72.1%

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

   if -1.64999999999999997e26 < x < -4.7000000000000001e-23 or 2.20000000000000006e-144 < x < 4.00000000000000035e-111

   1. Initial program 93.1%

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

      1. *-rgt-identity 93.1%

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

      2. fabs-sub 93.1%

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

      3. metadata-eval 93.1%

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

      4. metadata-eval 93.1%

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

      5. fabs-mul 93.1%

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

      6. metadata-eval 93.1%

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

      7. *-rgt-identity 93.1%

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

      8. associate-*l/ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. remove-double-neg 99.8%

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

      13. distribute-neg-in 99.8%

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

      14. sub-neg 99.8%

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

      15. neg-mul-1 99.8%

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

      16. associate-*l/ 99.5%

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

      17. remove-double-neg 99.5%

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

      18. distribute-lft-neg-in 99.5%

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

      19. distribute-rgt-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around inf 93.8%

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

   if -4.7000000000000001e-23 < x < 2.20000000000000006e-144 or 4.00000000000000035e-111 < x < 1.49999999999999994e-35

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 79.1%

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

   if 1.49999999999999994e-35 < x < 9.50000000000000002e182

   1. Initial program 89.8%

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

      1. *-rgt-identity 89.8%

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

      2. fabs-sub 89.8%

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

      3. metadata-eval 89.8%

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

      4. metadata-eval 89.8%

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

      5. fabs-mul 89.8%

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

      6. metadata-eval 89.8%

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

      7. *-rgt-identity 89.8%

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

      8. associate-*l/ 90.2%

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

      9. div-sub 95.2%

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

      10. sub-neg 95.2%

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

      11. +-commutative 95.2%

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

      12. remove-double-neg 95.2%

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

      13. distribute-neg-in 95.2%

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

      14. sub-neg 95.2%

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

      15. neg-mul-1 95.2%

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

      16. associate-*l/ 95.0%

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

      17. remove-double-neg 95.0%

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

      18. distribute-lft-neg-in 95.0%

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

      19. distribute-rgt-neg-in 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in z around inf 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l/ 64.5%

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

   6. Applied egg-rr 64.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-144}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+182}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-36}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* z (/ x y)))))
   (if (<= x -1.4e+26)
     t_0
     (if (<= x -7.8e-22)
       t_1
       (if (<= x 3.8e-36) (fabs (/ 4.0 y)) (if (<= x 3.8e+182) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((z * (x / y)));
	double tmp;
	if (x <= -1.4e+26) {
		tmp = t_0;
	} else if (x <= -7.8e-22) {
		tmp = t_1;
	} else if (x <= 3.8e-36) {
		tmp = fabs((4.0 / y));
	} else if (x <= 3.8e+182) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((z * (x / y)))
    if (x <= (-1.4d+26)) then
        tmp = t_0
    else if (x <= (-7.8d-22)) then
        tmp = t_1
    else if (x <= 3.8d-36) then
        tmp = abs((4.0d0 / y))
    else if (x <= 3.8d+182) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((z * (x / y)));
	double tmp;
	if (x <= -1.4e+26) {
		tmp = t_0;
	} else if (x <= -7.8e-22) {
		tmp = t_1;
	} else if (x <= 3.8e-36) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 3.8e+182) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((z * (x / y)))
	tmp = 0
	if x <= -1.4e+26:
		tmp = t_0
	elif x <= -7.8e-22:
		tmp = t_1
	elif x <= 3.8e-36:
		tmp = math.fabs((4.0 / y))
	elif x <= 3.8e+182:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -1.4e+26)
		tmp = t_0;
	elseif (x <= -7.8e-22)
		tmp = t_1;
	elseif (x <= 3.8e-36)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 3.8e+182)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((z * (x / y)));
	tmp = 0.0;
	if (x <= -1.4e+26)
		tmp = t_0;
	elseif (x <= -7.8e-22)
		tmp = t_1;
	elseif (x <= 3.8e-36)
		tmp = abs((4.0 / y));
	elseif (x <= 3.8e+182)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.4e+26], t$95$0, If[LessEqual[x, -7.8e-22], t$95$1, If[LessEqual[x, 3.8e-36], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.8e+182], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e26 or 3.80000000000000013e182 < x

   1. Initial program 84.8%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-lft-in 89.3%

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

      5. associate-*r/ 89.5%

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

      6. *-rgt-identity 89.5%

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

      7. mul-1-neg 89.5%

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

      8. distribute-rgt-neg-in 89.5%

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

      9. unsub-neg 89.5%

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

      10. *-lft-identity 89.5%

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

      11. associate-/l* 89.3%

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

      12. *-commutative 89.3%

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

      13. associate-/r/ 84.6%

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

      14. div-sub 99.8%

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

      15. associate-/r/ 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around 0 72.1%

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

   if -1.4e26 < x < -7.79999999999999996e-22 or 3.79999999999999971e-36 < x < 3.80000000000000013e182

   1. Initial program 91.7%

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

      1. *-rgt-identity 91.7%

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

      2. fabs-sub 91.7%

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

      3. metadata-eval 91.7%

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

      4. metadata-eval 91.7%

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

      5. fabs-mul 91.7%

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

      6. metadata-eval 91.7%

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

      7. *-rgt-identity 91.7%

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

      8. associate-*l/ 92.1%

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

      9. div-sub 96.2%

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

      10. sub-neg 96.2%

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

      11. +-commutative 96.2%

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

      12. remove-double-neg 96.2%

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

      13. distribute-neg-in 96.2%

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

      14. sub-neg 96.2%

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

      15. neg-mul-1 96.2%

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

      16. associate-*l/ 95.9%

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

      17. remove-double-neg 95.9%

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

      18. distribute-lft-neg-in 95.9%

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

      19. distribute-rgt-neg-in 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l/ 69.6%

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

   6. Applied egg-rr 69.6%

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

   if -7.79999999999999996e-22 < x < 3.79999999999999971e-36

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 76.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-36}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+182}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+32)
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+32) {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d+32)) then
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+32) {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e+32:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e+32)
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e+32)
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e+32], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000005e32

   1. Initial program 96.3%

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

      1. associate-*l/ 89.4%

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

      2. associate-/l* 99.8%

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

   3. Applied egg-rr 99.8%

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

   if -1.00000000000000005e32 < y

   1. Initial program 90.5%

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

      1. associate-*l/ 94.1%

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

      2. sub-div 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 5: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-26}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -1.95e+28)
     t_0
     (if (<= x -2.9e-26)
       (fabs (* x (/ z y)))
       (if (<= x 4.0) (fabs (/ 4.0 y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -1.95e+28) {
		tmp = t_0;
	} else if (x <= -2.9e-26) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if (x <= (-1.95d+28)) then
        tmp = t_0
    else if (x <= (-2.9d-26)) then
        tmp = abs((x * (z / y)))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -1.95e+28) {
		tmp = t_0;
	} else if (x <= -2.9e-26) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -1.95e+28:
		tmp = t_0
	elif x <= -2.9e-26:
		tmp = math.fabs((x * (z / y)))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.95e+28)
		tmp = t_0;
	elseif (x <= -2.9e-26)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.95e+28)
		tmp = t_0;
	elseif (x <= -2.9e-26)
		tmp = abs((x * (z / y)));
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.95e+28], t$95$0, If[LessEqual[x, -2.9e-26], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9499999999999999e28 or 4 < x

   1. Initial program 85.5%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. distribute-lft-in 90.1%

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

      5. associate-*r/ 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. mul-1-neg 90.3%

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

      8. distribute-rgt-neg-in 90.3%

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

      9. unsub-neg 90.3%

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

      10. *-lft-identity 90.3%

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

      11. associate-/l* 90.2%

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

      12. *-commutative 90.2%

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

      13. associate-/r/ 86.8%

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

      14. div-sub 99.5%

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

      15. associate-/r/ 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in z around 0 66.9%

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

   if -1.9499999999999999e28 < x < -2.8999999999999998e-26

   1. Initial program 99.4%

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

      1. *-rgt-identity 99.4%

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

      2. fabs-sub 99.4%

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

      3. metadata-eval 99.4%

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

      4. metadata-eval 99.4%

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

      5. fabs-mul 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-rgt-identity 99.4%

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

      8. associate-*l/ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. remove-double-neg 99.8%

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

      13. distribute-neg-in 99.8%

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

      14. sub-neg 99.8%

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

      15. neg-mul-1 99.8%

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

      16. associate-*l/ 99.4%

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

      17. remove-double-neg 99.4%

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

      18. distribute-lft-neg-in 99.4%

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

      19. distribute-rgt-neg-in 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around inf 90.8%

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

      1. associate-*l/ 82.2%

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

      2. *-commutative 82.2%

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

   6. Simplified 82.2%

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

   if -2.8999999999999998e-26 < x < 4

   1. Initial program 96.8%

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

   2. Taylor expanded in x around 0 73.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-26}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+106}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+84}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+106)
   (fabs (/ (* x z) y))
   (if (<= z 5.4e+84) (fabs (/ (- -4.0 x) y)) (fabs (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+106) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 5.4e+84) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.2d+106)) then
        tmp = abs(((x * z) / y))
    else if (z <= 5.4d+84) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+106) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 5.4e+84) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.2e+106:
		tmp = math.fabs(((x * z) / y))
	elif z <= 5.4e+84:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+106)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 5.4e+84)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.2e+106)
		tmp = abs(((x * z) / y));
	elseif (z <= 5.4e+84)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e+106], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 5.4e+84], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2e106

   1. Initial program 93.2%

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

      1. *-rgt-identity 93.2%

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

      2. fabs-sub 93.2%

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

      3. metadata-eval 93.2%

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

      4. metadata-eval 93.2%

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

      5. fabs-mul 93.2%

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

      6. metadata-eval 93.2%

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

      7. *-rgt-identity 93.2%

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

      8. associate-*l/ 95.3%

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

      9. div-sub 95.3%

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

      10. sub-neg 95.3%

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

      11. +-commutative 95.3%

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

      12. remove-double-neg 95.3%

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

      13. distribute-neg-in 95.3%

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

      14. sub-neg 95.3%

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

      15. neg-mul-1 95.3%

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

      16. associate-*l/ 95.3%

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

      17. remove-double-neg 95.3%

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

      18. distribute-lft-neg-in 95.3%

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

      19. distribute-rgt-neg-in 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in z around inf 86.5%

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

   if -1.2e106 < z < 5.4e84

   1. Initial program 93.9%

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

      1. *-rgt-identity 93.9%

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

      2. fabs-sub 93.9%

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

      3. metadata-eval 93.9%

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

      4. metadata-eval 93.9%

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

      5. fabs-mul 93.9%

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

      6. metadata-eval 93.9%

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

      7. *-rgt-identity 93.9%

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

      8. associate-*l/ 97.0%

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

      9. div-sub 99.5%

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

      10. sub-neg 99.5%

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

      11. +-commutative 99.5%

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

      12. remove-double-neg 99.5%

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

      13. distribute-neg-in 99.5%

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

      14. sub-neg 99.5%

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

      15. neg-mul-1 99.5%

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

      16. associate-*l/ 99.3%

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

      17. remove-double-neg 99.3%

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

      18. distribute-lft-neg-in 99.3%

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

      19. distribute-rgt-neg-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 92.1%

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

      1. associate-*r/ 92.1%

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

      2. distribute-lft-in 92.1%

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

      3. metadata-eval 92.1%

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

      4. neg-mul-1 92.1%

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

      5. sub-neg 92.1%

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

   6. Simplified 92.1%

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

   if 5.4e84 < z

   1. Initial program 83.0%

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

      1. *-rgt-identity 83.0%

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

      2. fabs-sub 83.0%

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

      3. metadata-eval 83.0%

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

      4. metadata-eval 83.0%

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

      5. fabs-mul 83.0%

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

      6. metadata-eval 83.0%

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

      7. *-rgt-identity 83.0%

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

      8. associate-*l/ 78.0%

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

      9. div-sub 92.2%

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

      10. sub-neg 92.2%

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

      11. +-commutative 92.2%

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

      12. remove-double-neg 92.2%

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

      13. distribute-neg-in 92.2%

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

      14. sub-neg 92.2%

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

      15. neg-mul-1 92.2%

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

      16. associate-*l/ 92.1%

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

      17. remove-double-neg 92.1%

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

      18. distribute-lft-neg-in 92.1%

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

      19. distribute-rgt-neg-in 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in z around inf 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-*l/ 89.4%

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

   6. Applied egg-rr 89.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+106}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+84}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. associate-*l/ 93.1%

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

   2. sub-div 97.4%

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

3. Applied egg-rr 97.4%

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

4. Final simplification 97.4%

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

Alternative 8: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. distribute-lft-in 90.6%

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

      5. associate-*r/ 90.8%

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

      6. *-rgt-identity 90.8%

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

      7. mul-1-neg 90.8%

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

      8. distribute-rgt-neg-in 90.8%

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

      9. unsub-neg 90.8%

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

      10. *-lft-identity 90.8%

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

      11. associate-/l* 90.6%

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

      12. *-commutative 90.6%

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

      13. associate-/r/ 87.4%

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

      14. div-sub 99.5%

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

      15. associate-/r/ 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in z around 0 63.9%

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

   if -10.5 < x < 4

   1. Initial program 96.9%

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

   2. Taylor expanded in x around 0 72.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 9: 40.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{4}{y}\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

2. Taylor expanded in x around 0 39.5%

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

3. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))