Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 83.9% → 97.8%

Time: 7.5s

Alternatives: 8

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (y + z)) / 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 = (x * (y + z)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

Python

def code(x, y, z):
	return (x * (y + z)) / z

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (y + z)) / 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 = (x * (y + z)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

Python

def code(x, y, z):
	return (x * (y + z)) / z

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;x\_m + \frac{1}{\frac{z}{x\_m \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{y}{z}, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.6e-48) (+ x_m (/ 1.0 (/ z (* x_m y)))) (fma x_m (/ y z) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.6e-48) {
		tmp = x_m + (1.0 / (z / (x_m * y)));
	} else {
		tmp = fma(x_m, (y / z), x_m);
	}
	return x_s * tmp;
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.6e-48)
		tmp = Float64(x_m + Float64(1.0 / Float64(z / Float64(x_m * y))));
	else
		tmp = fma(x_m, Float64(y / z), x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.6e-48], N[(x$95$m + N[(1.0 / N[(z / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y / z), $MachinePrecision] + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.5999999999999999e-48

   1. Initial program 86.4%

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

      1. remove-double-neg 86.4%

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

      2. distribute-lft-neg-out 86.4%

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

      3. *-commutative 86.4%

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

      4. distribute-lft-neg-in 86.4%

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

      5. associate-/l* 83.4%

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

      6. distribute-neg-in 83.4%

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

      7. unsub-neg 83.4%

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

      8. div-sub 80.3%

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

      9. distribute-frac-neg 80.3%

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

      10. associate-/r/ 74.9%

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

      11. distribute-rgt-neg-out 74.9%

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

      12. remove-double-neg 74.9%

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

      13. associate-/r/ 90.6%

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

      14. *-inverses 90.6%

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

      15. *-lft-identity 90.6%

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

      16. *-commutative 90.6%

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

      17. fma-neg 90.6%

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

      18. remove-double-neg 90.6%

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

   3. Simplified 90.6%

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

      1. fma-udef 90.6%

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

   6. Applied egg-rr 90.6%

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

      1. associate-*r/ 92.3%

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

      2. clear-num 92.3%

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

   8. Applied egg-rr 92.3%

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

   if 1.5999999999999999e-48 < x

   1. Initial program 70.9%

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

      1. remove-double-neg 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. *-commutative 70.9%

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

      4. distribute-lft-neg-in 70.9%

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

      5. associate-/l* 94.3%

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

      6. distribute-neg-in 94.3%

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

      7. unsub-neg 94.3%

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

      8. div-sub 89.3%

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

      9. distribute-frac-neg 89.3%

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

      10. associate-/r/ 90.0%

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

      11. distribute-rgt-neg-out 90.0%

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

      12. remove-double-neg 90.0%

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

      13. associate-/r/ 99.8%

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

      14. *-inverses 99.8%

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

      15. *-lft-identity 99.8%

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

      16. *-commutative 99.8%

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

      17. fma-neg 99.9%

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

      18. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-61} \lor \neg \left(z \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.45e-61) (not (<= z 2.5e-7)))
    (+ x_m (* x_m (/ y z)))
    (* (+ z y) (/ x_m z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.45e-61) || !(z <= 2.5e-7)) {
		tmp = x_m + (x_m * (y / z));
	} else {
		tmp = (z + y) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.45d-61)) .or. (.not. (z <= 2.5d-7))) then
        tmp = x_m + (x_m * (y / z))
    else
        tmp = (z + y) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.45e-61) || !(z <= 2.5e-7)) {
		tmp = x_m + (x_m * (y / z));
	} else {
		tmp = (z + y) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (z <= -1.45e-61) or not (z <= 2.5e-7):
		tmp = x_m + (x_m * (y / z))
	else:
		tmp = (z + y) * (x_m / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.45e-61) || !(z <= 2.5e-7))
		tmp = Float64(x_m + Float64(x_m * Float64(y / z)));
	else
		tmp = Float64(Float64(z + y) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -1.45e-61) || ~((z <= 2.5e-7)))
		tmp = x_m + (x_m * (y / z));
	else
		tmp = (z + y) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[z, -1.45e-61], N[Not[LessEqual[z, 2.5e-7]], $MachinePrecision]], N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e-61 or 2.49999999999999989e-7 < z

   1. Initial program 78.0%

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

      1. remove-double-neg 78.0%

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

      2. distribute-lft-neg-out 78.0%

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

      3. *-commutative 78.0%

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

      4. distribute-lft-neg-in 78.0%

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

      5. associate-/l* 79.1%

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

      6. distribute-neg-in 79.1%

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

      7. unsub-neg 79.1%

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

      8. div-sub 78.4%

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

      9. distribute-frac-neg 78.4%

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

      10. associate-/r/ 83.5%

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

      11. distribute-rgt-neg-out 83.5%

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

      12. remove-double-neg 83.5%

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

      13. associate-/r/ 99.8%

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

      14. *-inverses 99.8%

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

      15. *-lft-identity 99.8%

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

      16. *-commutative 99.8%

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

      17. fma-neg 99.8%

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

      18. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -1.45e-61 < z < 2.49999999999999989e-7

   1. Initial program 88.8%

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

      1. associate-*l/ 93.9%

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

      2. *-commutative 93.9%

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

   3. Simplified 93.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-61} \lor \neg \left(z \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+164}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+146}:\\ \;\;\;\;\left(z + y\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -4.6e+164) x_m (if (<= z 2.4e+146) (* (+ z y) (/ x_m z)) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -4.6e+164) {
		tmp = x_m;
	} else if (z <= 2.4e+146) {
		tmp = (z + y) * (x_m / z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.6d+164)) then
        tmp = x_m
    else if (z <= 2.4d+146) then
        tmp = (z + y) * (x_m / z)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -4.6e+164) {
		tmp = x_m;
	} else if (z <= 2.4e+146) {
		tmp = (z + y) * (x_m / z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -4.6e+164:
		tmp = x_m
	elif z <= 2.4e+146:
		tmp = (z + y) * (x_m / z)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -4.6e+164)
		tmp = x_m;
	elseif (z <= 2.4e+146)
		tmp = Float64(Float64(z + y) * Float64(x_m / z));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -4.6e+164)
		tmp = x_m;
	elseif (z <= 2.4e+146)
		tmp = (z + y) * (x_m / z);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -4.6e+164], x$95$m, If[LessEqual[z, 2.4e+146], N[(N[(z + y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.5999999999999999e164 or 2.4000000000000002e146 < z

   1. Initial program 66.3%

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

      1. associate-*l/ 64.2%

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

      2. *-commutative 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in y around 0 86.2%

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

   if -4.5999999999999999e164 < z < 2.4000000000000002e146

   1. Initial program 88.1%

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

      1. associate-*l/ 93.6%

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

      2. *-commutative 93.6%

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

   3. Simplified 93.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+164}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+146}:\\ \;\;\;\;\left(z + y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= z -7.8e-19) x_m (if (<= z 2.75e+63) (* y (/ x_m z)) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -7.8e-19) {
		tmp = x_m;
	} else if (z <= 2.75e+63) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7.8d-19)) then
        tmp = x_m
    else if (z <= 2.75d+63) then
        tmp = y * (x_m / z)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -7.8e-19) {
		tmp = x_m;
	} else if (z <= 2.75e+63) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -7.8e-19:
		tmp = x_m
	elif z <= 2.75e+63:
		tmp = y * (x_m / z)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -7.8e-19)
		tmp = x_m;
	elseif (z <= 2.75e+63)
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -7.8e-19)
		tmp = x_m;
	elseif (z <= 2.75e+63)
		tmp = y * (x_m / z);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -7.8e-19], x$95$m, If[LessEqual[z, 2.75e+63], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999999e-19 or 2.75000000000000002e63 < z

   1. Initial program 74.2%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in y around 0 74.0%

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

   if -7.7999999999999999e-19 < z < 2.75000000000000002e63

   1. Initial program 90.2%

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

      1. associate-*l/ 94.1%

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

      2. *-commutative 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 78.2%

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

      1. associate-*l/ 78.2%

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

      2. *-commutative 78.2%

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

   7. Simplified 78.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= z -7.8e-19) x_m (if (<= z 1.45e+64) (/ y (/ z x_m)) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -7.8e-19) {
		tmp = x_m;
	} else if (z <= 1.45e+64) {
		tmp = y / (z / x_m);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7.8d-19)) then
        tmp = x_m
    else if (z <= 1.45d+64) then
        tmp = y / (z / x_m)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -7.8e-19) {
		tmp = x_m;
	} else if (z <= 1.45e+64) {
		tmp = y / (z / x_m);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -7.8e-19:
		tmp = x_m
	elif z <= 1.45e+64:
		tmp = y / (z / x_m)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -7.8e-19)
		tmp = x_m;
	elseif (z <= 1.45e+64)
		tmp = Float64(y / Float64(z / x_m));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -7.8e-19)
		tmp = x_m;
	elseif (z <= 1.45e+64)
		tmp = y / (z / x_m);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -7.8e-19], x$95$m, If[LessEqual[z, 1.45e+64], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999999e-19 or 1.44999999999999997e64 < z

   1. Initial program 74.2%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in y around 0 74.0%

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

   if -7.7999999999999999e-19 < z < 1.44999999999999997e64

   1. Initial program 90.2%

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

      1. associate-*l/ 94.1%

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

      2. *-commutative 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 78.2%

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

      1. associate-/l* 70.0%

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

   7. Simplified 70.0%

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

      1. associate-/l* 78.2%

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

      2. associate-*r/ 70.0%

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

      3. *-commutative 70.0%

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

   9. Applied egg-rr 70.0%

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

       1. associate-/r/ 78.8%

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

   11. Applied egg-rr 78.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-46}:\\ \;\;\;\;x\_m + \frac{1}{\frac{z}{x\_m \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e-46) (+ x_m (/ 1.0 (/ z (* x_m y)))) (+ x_m (* x_m (/ y z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1e-46) {
		tmp = x_m + (1.0 / (z / (x_m * y)));
	} else {
		tmp = x_m + (x_m * (y / z));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1d-46) then
        tmp = x_m + (1.0d0 / (z / (x_m * y)))
    else
        tmp = x_m + (x_m * (y / z))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1e-46) {
		tmp = x_m + (1.0 / (z / (x_m * y)));
	} else {
		tmp = x_m + (x_m * (y / z));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 1e-46:
		tmp = x_m + (1.0 / (z / (x_m * y)))
	else:
		tmp = x_m + (x_m * (y / z))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1e-46)
		tmp = Float64(x_m + Float64(1.0 / Float64(z / Float64(x_m * y))));
	else
		tmp = Float64(x_m + Float64(x_m * Float64(y / z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 1e-46)
		tmp = x_m + (1.0 / (z / (x_m * y)));
	else
		tmp = x_m + (x_m * (y / z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1e-46], N[(x$95$m + N[(1.0 / N[(z / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000002e-46

   1. Initial program 86.4%

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

      1. remove-double-neg 86.4%

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

      2. distribute-lft-neg-out 86.4%

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

      3. *-commutative 86.4%

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

      4. distribute-lft-neg-in 86.4%

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

      5. associate-/l* 83.4%

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

      6. distribute-neg-in 83.4%

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

      7. unsub-neg 83.4%

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

      8. div-sub 80.3%

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

      9. distribute-frac-neg 80.3%

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

      10. associate-/r/ 74.9%

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

      11. distribute-rgt-neg-out 74.9%

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

      12. remove-double-neg 74.9%

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

      13. associate-/r/ 90.6%

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

      14. *-inverses 90.6%

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

      15. *-lft-identity 90.6%

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

      16. *-commutative 90.6%

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

      17. fma-neg 90.6%

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

      18. remove-double-neg 90.6%

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

   3. Simplified 90.6%

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

      1. fma-udef 90.6%

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

   6. Applied egg-rr 90.6%

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

      1. associate-*r/ 92.3%

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

      2. clear-num 92.3%

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

   8. Applied egg-rr 92.3%

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

   if 1.00000000000000002e-46 < x

   1. Initial program 70.9%

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

      1. remove-double-neg 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. *-commutative 70.9%

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

      4. distribute-lft-neg-in 70.9%

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

      5. associate-/l* 94.3%

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

      6. distribute-neg-in 94.3%

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

      7. unsub-neg 94.3%

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

      8. div-sub 89.3%

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

      9. distribute-frac-neg 89.3%

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

      10. associate-/r/ 90.0%

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

      11. distribute-rgt-neg-out 90.0%

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

      12. remove-double-neg 90.0%

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

      13. associate-/r/ 99.8%

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

      14. *-inverses 99.8%

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

      15. *-lft-identity 99.8%

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

      16. *-commutative 99.8%

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

      17. fma-neg 99.9%

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

      18. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-46}:\\ \;\;\;\;x + \frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{-127}:\\ \;\;\;\;x\_m + \frac{y}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m + x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1e-127) (+ x_m (/ y (/ z x_m))) (+ x_m (* x_m (/ y z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.1e-127) {
		tmp = x_m + (y / (z / x_m));
	} else {
		tmp = x_m + (x_m * (y / z));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2.1d-127) then
        tmp = x_m + (y / (z / x_m))
    else
        tmp = x_m + (x_m * (y / z))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.1e-127) {
		tmp = x_m + (y / (z / x_m));
	} else {
		tmp = x_m + (x_m * (y / z));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2.1e-127:
		tmp = x_m + (y / (z / x_m))
	else:
		tmp = x_m + (x_m * (y / z))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.1e-127)
		tmp = Float64(x_m + Float64(y / Float64(z / x_m)));
	else
		tmp = Float64(x_m + Float64(x_m * Float64(y / z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.1e-127)
		tmp = x_m + (y / (z / x_m));
	else
		tmp = x_m + (x_m * (y / z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.1e-127], N[(x$95$m + N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.1000000000000001e-127

   1. Initial program 85.3%

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

      1. remove-double-neg 85.3%

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

      2. distribute-lft-neg-out 85.3%

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

      3. *-commutative 85.3%

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

      4. distribute-lft-neg-in 85.3%

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

      5. associate-/l* 83.7%

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

      6. distribute-neg-in 83.7%

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

      7. unsub-neg 83.7%

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

      8. div-sub 80.4%

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

      9. distribute-frac-neg 80.4%

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

      10. associate-/r/ 75.1%

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

      11. distribute-rgt-neg-out 75.1%

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

      12. remove-double-neg 75.1%

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

      13. associate-/r/ 90.4%

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

      14. *-inverses 90.4%

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

      15. *-lft-identity 90.4%

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

      16. *-commutative 90.4%

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

      17. fma-neg 90.4%

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

      18. remove-double-neg 90.4%

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

   3. Simplified 90.4%

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

      1. fma-udef 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. associate-*r/ 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-/l* 94.1%

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

   8. Applied egg-rr 94.1%

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

   if 2.1000000000000001e-127 < x

   1. Initial program 76.3%

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

      1. remove-double-neg 76.3%

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

      2. distribute-lft-neg-out 76.3%

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

      3. *-commutative 76.3%

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

      4. distribute-lft-neg-in 76.3%

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

      5. associate-/l* 91.5%

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

      6. distribute-neg-in 91.5%

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

      7. unsub-neg 91.5%

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

      8. div-sub 87.4%

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

      9. distribute-frac-neg 87.4%

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

      10. associate-/r/ 86.8%

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

      11. distribute-rgt-neg-out 86.8%

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

      12. remove-double-neg 86.8%

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

      13. associate-/r/ 98.6%

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

      14. *-inverses 98.6%

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

      15. *-lft-identity 98.6%

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

      16. *-commutative 98.6%

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

      17. fma-neg 98.6%

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

      18. remove-double-neg 98.6%

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

   3. Simplified 98.6%

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

      1. fma-udef 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-127}:\\ \;\;\;\;x + \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. associate-*l/ 86.3%

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

   2. *-commutative 86.3%

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

3. Simplified 86.3%

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

5. Taylor expanded in y around 0 44.5%

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

6. Final simplification 44.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y + z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ x (/ z (+ y z))))

C

double code(double x, double y, double z) {
	return x / (z / (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 = x / (z / (y + z))
end function

Java

public static double code(double x, double y, double z) {
	return x / (z / (y + z));
}

Python

def code(x, y, z):
	return x / (z / (y + z))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))