Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.1% → 96.5%

Time: 4.6s

Alternatives: 10

Speedup: 1.0×

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: 84.1% 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: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x_m \cdot \left(y + z\right)}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -5 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x_m + \frac{x_m}{\frac{z}{y}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (+ y z)) z)))
   (* x_s (if (<= t_0 -5e+124) t_0 (+ x_m (/ x_m (/ z y)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y + z)) / z;
	double tmp;
	if (t_0 <= -5e+124) {
		tmp = t_0;
	} else {
		tmp = x_m + (x_m / (z / y));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (y + z)) / z
    if (t_0 <= (-5d+124)) then
        tmp = t_0
    else
        tmp = x_m + (x_m / (z / y))
    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 t_0 = (x_m * (y + z)) / z;
	double tmp;
	if (t_0 <= -5e+124) {
		tmp = t_0;
	} else {
		tmp = x_m + (x_m / (z / y));
	}
	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):
	t_0 = (x_m * (y + z)) / z
	tmp = 0
	if t_0 <= -5e+124:
		tmp = t_0
	else:
		tmp = x_m + (x_m / (z / y))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(y + z)) / z)
	tmp = 0.0
	if (t_0 <= -5e+124)
		tmp = t_0;
	else
		tmp = Float64(x_m + Float64(x_m / Float64(z / y)));
	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)
	t_0 = (x_m * (y + z)) / z;
	tmp = 0.0;
	if (t_0 <= -5e+124)
		tmp = t_0;
	else
		tmp = x_m + (x_m / (z / y));
	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_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e+124], t$95$0, N[(x$95$m + N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -4.9999999999999996e124

   1. Initial program 79.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   if -4.9999999999999996e124 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 86.5%

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

      1. remove-double-neg 86.5%

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

      2. distribute-lft-neg-out 86.5%

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

      3. *-commutative 86.5%

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

      4. distribute-lft-neg-in 86.5%

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

      5. associate-/l* 84.2%

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

      6. distribute-neg-in 84.2%

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

      7. unsub-neg 84.2%

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

      8. div-sub 81.5%

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

      9. distribute-frac-neg 81.5%

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

      10. associate-/r/ 82.2%

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

      11. distribute-rgt-neg-out 82.2%

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

      12. remove-double-neg 82.2%

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

      13. associate-/r/ 97.5%

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

      14. *-inverses 97.5%

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

      15. *-lft-identity 97.5%

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

      16. *-commutative 97.5%

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

      17. fma-neg 97.4%

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

      18. remove-double-neg 97.4%

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

   3. Simplified 97.4%

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

      1. fma-udef 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. associate-*r/ 93.2%

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

      2. associate-/l* 97.9%

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;x_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-12}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-249}:\\ \;\;\;\;\frac{x_m}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+38}:\\ \;\;\;\;\frac{y}{\frac{z}{x_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

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 -1.85e+101)
    x_m
    (if (<= z -1.4e+43)
      (* x_m (/ y z))
      (if (<= z -5.2e-12)
        x_m
        (if (<= z 8.6e-249)
          (/ x_m (/ z y))
          (if (<= z 2.85e+38) (/ 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 <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -1.4e+43) {
		tmp = x_m * (y / z);
	} else if (z <= -5.2e-12) {
		tmp = x_m;
	} else if (z <= 8.6e-249) {
		tmp = x_m / (z / y);
	} else if (z <= 2.85e+38) {
		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 <= (-1.85d+101)) then
        tmp = x_m
    else if (z <= (-1.4d+43)) then
        tmp = x_m * (y / z)
    else if (z <= (-5.2d-12)) then
        tmp = x_m
    else if (z <= 8.6d-249) then
        tmp = x_m / (z / y)
    else if (z <= 2.85d+38) 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 <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -1.4e+43) {
		tmp = x_m * (y / z);
	} else if (z <= -5.2e-12) {
		tmp = x_m;
	} else if (z <= 8.6e-249) {
		tmp = x_m / (z / y);
	} else if (z <= 2.85e+38) {
		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 <= -1.85e+101:
		tmp = x_m
	elif z <= -1.4e+43:
		tmp = x_m * (y / z)
	elif z <= -5.2e-12:
		tmp = x_m
	elif z <= 8.6e-249:
		tmp = x_m / (z / y)
	elif z <= 2.85e+38:
		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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -1.4e+43)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= -5.2e-12)
		tmp = x_m;
	elseif (z <= 8.6e-249)
		tmp = Float64(x_m / Float64(z / y));
	elseif (z <= 2.85e+38)
		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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -1.4e+43)
		tmp = x_m * (y / z);
	elseif (z <= -5.2e-12)
		tmp = x_m;
	elseif (z <= 8.6e-249)
		tmp = x_m / (z / y);
	elseif (z <= 2.85e+38)
		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, -1.85e+101], x$95$m, If[LessEqual[z, -1.4e+43], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5.2e-12], x$95$m, If[LessEqual[z, 8.6e-249], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.85e+38], N[(y / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], x$95$m]]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -1.8499999999999999e101 or -1.40000000000000009e43 < z < -5.19999999999999965e-12 or 2.8499999999999999e38 < z

   1. Initial program 71.5%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in y around 0 79.5%

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

   if -1.8499999999999999e101 < z < -1.40000000000000009e43

   1. Initial program 89.7%

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

      1. associate-*l/ 88.9%

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

      2. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 57.9%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

   if -5.19999999999999965e-12 < z < 8.6000000000000003e-249

   1. Initial program 92.9%

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

      1. associate-*l/ 88.9%

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

      2. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 77.7%

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

      1. associate-*r/ 83.4%

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

   7. Simplified 83.4%

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

      1. associate-*r/ 94.3%

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

      2. associate-/l* 99.9%

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

   9. Applied egg-rr 83.4%

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

   if 8.6000000000000003e-249 < z < 2.8499999999999999e38

   1. Initial program 95.6%

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

      1. associate-*l/ 94.3%

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

      2. *-commutative 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in y around inf 70.8%

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

      1. associate-*r/ 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*r/ 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-/l* 70.8%

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

   9. Applied egg-rr 70.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-249}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+38}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(y + z\right) \cdot \frac{x_m}{z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+153}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-283}:\\ \;\;\;\;x_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+201}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ y z) (/ x_m z))))
   (*
    x_s
    (if (<= z -5.6e+153)
      x_m
      (if (<= z -3.6e-200)
        t_0
        (if (<= z 5.5e-283) (* x_m (/ y z)) (if (<= z 1.7e+201) t_0 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 t_0 = (y + z) * (x_m / z);
	double tmp;
	if (z <= -5.6e+153) {
		tmp = x_m;
	} else if (z <= -3.6e-200) {
		tmp = t_0;
	} else if (z <= 5.5e-283) {
		tmp = x_m * (y / z);
	} else if (z <= 1.7e+201) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y + z) * (x_m / z)
    if (z <= (-5.6d+153)) then
        tmp = x_m
    else if (z <= (-3.6d-200)) then
        tmp = t_0
    else if (z <= 5.5d-283) then
        tmp = x_m * (y / z)
    else if (z <= 1.7d+201) then
        tmp = t_0
    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 t_0 = (y + z) * (x_m / z);
	double tmp;
	if (z <= -5.6e+153) {
		tmp = x_m;
	} else if (z <= -3.6e-200) {
		tmp = t_0;
	} else if (z <= 5.5e-283) {
		tmp = x_m * (y / z);
	} else if (z <= 1.7e+201) {
		tmp = t_0;
	} 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):
	t_0 = (y + z) * (x_m / z)
	tmp = 0
	if z <= -5.6e+153:
		tmp = x_m
	elif z <= -3.6e-200:
		tmp = t_0
	elif z <= 5.5e-283:
		tmp = x_m * (y / z)
	elif z <= 1.7e+201:
		tmp = t_0
	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)
	t_0 = Float64(Float64(y + z) * Float64(x_m / z))
	tmp = 0.0
	if (z <= -5.6e+153)
		tmp = x_m;
	elseif (z <= -3.6e-200)
		tmp = t_0;
	elseif (z <= 5.5e-283)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= 1.7e+201)
		tmp = t_0;
	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)
	t_0 = (y + z) * (x_m / z);
	tmp = 0.0;
	if (z <= -5.6e+153)
		tmp = x_m;
	elseif (z <= -3.6e-200)
		tmp = t_0;
	elseif (z <= 5.5e-283)
		tmp = x_m * (y / z);
	elseif (z <= 1.7e+201)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y + z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -5.6e+153], x$95$m, If[LessEqual[z, -3.6e-200], t$95$0, If[LessEqual[z, 5.5e-283], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e+201], t$95$0, x$95$m]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5999999999999997e153 or 1.7e201 < z

   1. Initial program 56.2%

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

      1. associate-*l/ 63.5%

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

      2. *-commutative 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in y around 0 94.2%

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

   if -5.5999999999999997e153 < z < -3.6000000000000002e-200 or 5.49999999999999953e-283 < z < 1.7e201

   1. Initial program 92.5%

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

   if -3.6000000000000002e-200 < z < 5.49999999999999953e-283

   1. Initial program 85.8%

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

      1. associate-*l/ 75.5%

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

      2. *-commutative 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in y around inf 85.5%

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

      1. associate-*r/ 96.1%

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

   7. Simplified 96.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+153}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-200}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-283}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+201}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;x_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-22}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x_m}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

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 -1.85e+101)
    x_m
    (if (<= z -1.55e+43)
      (* x_m (/ y z))
      (if (<= z -4.5e-22) x_m (if (<= z 6.1e+38) (/ x_m (/ z y)) 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 <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -1.55e+43) {
		tmp = x_m * (y / z);
	} else if (z <= -4.5e-22) {
		tmp = x_m;
	} else if (z <= 6.1e+38) {
		tmp = x_m / (z / y);
	} 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 <= (-1.85d+101)) then
        tmp = x_m
    else if (z <= (-1.55d+43)) then
        tmp = x_m * (y / z)
    else if (z <= (-4.5d-22)) then
        tmp = x_m
    else if (z <= 6.1d+38) then
        tmp = x_m / (z / y)
    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 <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -1.55e+43) {
		tmp = x_m * (y / z);
	} else if (z <= -4.5e-22) {
		tmp = x_m;
	} else if (z <= 6.1e+38) {
		tmp = x_m / (z / y);
	} 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 <= -1.85e+101:
		tmp = x_m
	elif z <= -1.55e+43:
		tmp = x_m * (y / z)
	elif z <= -4.5e-22:
		tmp = x_m
	elif z <= 6.1e+38:
		tmp = x_m / (z / y)
	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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -1.55e+43)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= -4.5e-22)
		tmp = x_m;
	elseif (z <= 6.1e+38)
		tmp = Float64(x_m / Float64(z / y));
	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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -1.55e+43)
		tmp = x_m * (y / z);
	elseif (z <= -4.5e-22)
		tmp = x_m;
	elseif (z <= 6.1e+38)
		tmp = x_m / (z / y);
	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, -1.85e+101], x$95$m, If[LessEqual[z, -1.55e+43], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4.5e-22], x$95$m, If[LessEqual[z, 6.1e+38], N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision], x$95$m]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.8499999999999999e101 or -1.5500000000000001e43 < z < -4.49999999999999987e-22 or 6.0999999999999999e38 < z

   1. Initial program 71.5%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in y around 0 79.5%

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

   if -1.8499999999999999e101 < z < -1.5500000000000001e43

   1. Initial program 89.7%

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

      1. associate-*l/ 88.9%

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

      2. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 57.9%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

   if -4.49999999999999987e-22 < z < 6.0999999999999999e38

   1. Initial program 94.2%

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

      1. associate-*l/ 91.6%

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

      2. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in y around inf 74.3%

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

      1. associate-*r/ 73.0%

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

   7. Simplified 73.0%

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

      1. associate-*r/ 96.2%

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

      2. associate-/l* 96.2%

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

   9. Applied egg-rr 74.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-22}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{+42}:\\ \;\;\;\;x_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-21}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+38}:\\ \;\;\;\;\frac{x_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

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 -1.85e+101)
    x_m
    (if (<= z -4.6e+42)
      (* x_m (/ y z))
      (if (<= z -3.6e-21) x_m (if (<= z 3.15e+38) (/ (* 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 (z <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -4.6e+42) {
		tmp = x_m * (y / z);
	} else if (z <= -3.6e-21) {
		tmp = x_m;
	} else if (z <= 3.15e+38) {
		tmp = (x_m * y) / 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 <= (-1.85d+101)) then
        tmp = x_m
    else if (z <= (-4.6d+42)) then
        tmp = x_m * (y / z)
    else if (z <= (-3.6d-21)) then
        tmp = x_m
    else if (z <= 3.15d+38) then
        tmp = (x_m * y) / 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 <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -4.6e+42) {
		tmp = x_m * (y / z);
	} else if (z <= -3.6e-21) {
		tmp = x_m;
	} else if (z <= 3.15e+38) {
		tmp = (x_m * y) / 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 <= -1.85e+101:
		tmp = x_m
	elif z <= -4.6e+42:
		tmp = x_m * (y / z)
	elif z <= -3.6e-21:
		tmp = x_m
	elif z <= 3.15e+38:
		tmp = (x_m * y) / 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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -4.6e+42)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= -3.6e-21)
		tmp = x_m;
	elseif (z <= 3.15e+38)
		tmp = Float64(Float64(x_m * y) / 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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -4.6e+42)
		tmp = x_m * (y / z);
	elseif (z <= -3.6e-21)
		tmp = x_m;
	elseif (z <= 3.15e+38)
		tmp = (x_m * y) / 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, -1.85e+101], x$95$m, If[LessEqual[z, -4.6e+42], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.6e-21], x$95$m, If[LessEqual[z, 3.15e+38], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], x$95$m]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.8499999999999999e101 or -4.6e42 < z < -3.59999999999999989e-21 or 3.15000000000000001e38 < z

   1. Initial program 71.5%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in y around 0 79.5%

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

   if -1.8499999999999999e101 < z < -4.6e42

   1. Initial program 89.7%

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

      1. associate-*l/ 88.9%

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

      2. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 57.9%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

   if -3.59999999999999989e-21 < z < 3.15000000000000001e38

   1. Initial program 94.2%

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

      1. associate-*l/ 91.6%

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

      2. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in y around inf 74.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+38}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x_m\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;x_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x_m \cdot z}{z}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

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 -1.85e+101)
    x_m
    (if (<= z -5.4e+42)
      (* x_m (/ y z))
      (if (<= z -6.5e-17)
        (/ (* x_m z) z)
        (if (<= z 4.1e+38) (/ (* 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 (z <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -5.4e+42) {
		tmp = x_m * (y / z);
	} else if (z <= -6.5e-17) {
		tmp = (x_m * z) / z;
	} else if (z <= 4.1e+38) {
		tmp = (x_m * y) / 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 <= (-1.85d+101)) then
        tmp = x_m
    else if (z <= (-5.4d+42)) then
        tmp = x_m * (y / z)
    else if (z <= (-6.5d-17)) then
        tmp = (x_m * z) / z
    else if (z <= 4.1d+38) then
        tmp = (x_m * y) / 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 <= -1.85e+101) {
		tmp = x_m;
	} else if (z <= -5.4e+42) {
		tmp = x_m * (y / z);
	} else if (z <= -6.5e-17) {
		tmp = (x_m * z) / z;
	} else if (z <= 4.1e+38) {
		tmp = (x_m * y) / 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 <= -1.85e+101:
		tmp = x_m
	elif z <= -5.4e+42:
		tmp = x_m * (y / z)
	elif z <= -6.5e-17:
		tmp = (x_m * z) / z
	elif z <= 4.1e+38:
		tmp = (x_m * y) / 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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -5.4e+42)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= -6.5e-17)
		tmp = Float64(Float64(x_m * z) / z);
	elseif (z <= 4.1e+38)
		tmp = Float64(Float64(x_m * y) / 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 <= -1.85e+101)
		tmp = x_m;
	elseif (z <= -5.4e+42)
		tmp = x_m * (y / z);
	elseif (z <= -6.5e-17)
		tmp = (x_m * z) / z;
	elseif (z <= 4.1e+38)
		tmp = (x_m * y) / 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, -1.85e+101], x$95$m, If[LessEqual[z, -5.4e+42], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -6.5e-17], N[(N[(x$95$m * z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 4.1e+38], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], x$95$m]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -1.8499999999999999e101 or 4.1000000000000003e38 < z

   1. Initial program 68.4%

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

      1. associate-*l/ 75.4%

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

      2. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in y around 0 79.4%

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

   if -1.8499999999999999e101 < z < -5.4000000000000001e42

   1. Initial program 89.7%

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

      1. associate-*l/ 88.9%

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

      2. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around inf 57.9%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

   if -5.4000000000000001e42 < z < -6.4999999999999996e-17

   1. Initial program 99.8%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.4%

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

   if -6.4999999999999996e-17 < z < 4.1000000000000003e38

   1. Initial program 94.2%

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

      1. associate-*l/ 91.6%

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

      2. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in y around inf 74.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x \cdot z}{z}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-81} \lor \neg \left(y \leq 2.7 \cdot 10^{-111}\right):\\ \;\;\;\;x_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (or (<= y -1.35e-81) (not (<= y 2.7e-111))) (* 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 ((y <= -1.35e-81) || !(y <= 2.7e-111)) {
		tmp = x_m * (y / 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 ((y <= (-1.35d-81)) .or. (.not. (y <= 2.7d-111))) then
        tmp = x_m * (y / 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 ((y <= -1.35e-81) || !(y <= 2.7e-111)) {
		tmp = x_m * (y / 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 (y <= -1.35e-81) or not (y <= 2.7e-111):
		tmp = x_m * (y / 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 ((y <= -1.35e-81) || !(y <= 2.7e-111))
		tmp = Float64(x_m * Float64(y / 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 ((y <= -1.35e-81) || ~((y <= 2.7e-111)))
		tmp = x_m * (y / 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[Or[LessEqual[y, -1.35e-81], N[Not[LessEqual[y, 2.7e-111]], $MachinePrecision]], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.34999999999999995e-81 or 2.69999999999999989e-111 < y

   1. Initial program 88.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 inf 70.7%

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

      1. associate-*r/ 70.2%

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

   7. Simplified 70.2%

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

   if -1.34999999999999995e-81 < y < 2.69999999999999989e-111

   1. Initial program 78.1%

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

      1. associate-*l/ 85.2%

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

      2. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in y around 0 82.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-81} \lor \neg \left(y \leq 2.7 \cdot 10^{-111}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + x_m \cdot \frac{y}{z}\right) \end{array} \]

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 (* 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) {
	return x_s * (x_m + (x_m * (y / z)));
}

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 + (x_m * (y / z)))
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 + (x_m * (y / z)));
}

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 + (x_m * (y / z)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m + Float64(x_m * Float64(y / z))))
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 + (x_m * (y / z)));
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 * N[(x$95$m + N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

   1. remove-double-neg 84.8%

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

   2. distribute-lft-neg-out 84.8%

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

   3. *-commutative 84.8%

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

   4. distribute-lft-neg-in 84.8%

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

   5. associate-/l* 86.2%

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

   6. distribute-neg-in 86.2%

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

   7. unsub-neg 86.2%

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

   8. div-sub 80.6%

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

   9. distribute-frac-neg 80.6%

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

   10. associate-/r/ 80.5%

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

   11. distribute-rgt-neg-out 80.5%

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

   12. remove-double-neg 80.5%

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

   13. associate-/r/ 97.3%

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

   14. *-inverses 97.3%

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

   15. *-lft-identity 97.3%

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

   16. *-commutative 97.3%

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

   17. fma-neg 97.3%

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

   18. remove-double-neg 97.3%

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

3. Simplified 97.3%

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

   1. fma-udef 97.3%

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

6. Applied egg-rr 97.3%

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

7. Final simplification 97.3%

   \[\leadsto x + x \cdot \frac{y}{z} \]
8. Add Preprocessing

Alternative 9: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + \frac{x_m}{\frac{z}{y}}\right) \end{array} \]

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 (/ x_m (/ z y)))))

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 + (x_m / (z / y)));
}

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 + (x_m / (z / y)))
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 + (x_m / (z / y)));
}

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 + (x_m / (z / y)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m + Float64(x_m / Float64(z / y))))
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 + (x_m / (z / y)));
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 * N[(x$95$m + N[(x$95$m / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

   1. remove-double-neg 84.8%

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

   2. distribute-lft-neg-out 84.8%

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

   3. *-commutative 84.8%

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

   4. distribute-lft-neg-in 84.8%

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

   5. associate-/l* 86.2%

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

   6. distribute-neg-in 86.2%

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

   7. unsub-neg 86.2%

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

   8. div-sub 80.6%

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

   9. distribute-frac-neg 80.6%

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

   10. associate-/r/ 80.5%

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

   11. distribute-rgt-neg-out 80.5%

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

   12. remove-double-neg 80.5%

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

   13. associate-/r/ 97.3%

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

   14. *-inverses 97.3%

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

   15. *-lft-identity 97.3%

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

   16. *-commutative 97.3%

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

   17. fma-neg 97.3%

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

   18. remove-double-neg 97.3%

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

3. Simplified 97.3%

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

   1. fma-udef 97.3%

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

6. Applied egg-rr 97.3%

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

   1. associate-*r/ 94.2%

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

   2. associate-/l* 98.0%

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

8. Applied egg-rr 98.0%

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

9. Final simplification 98.0%

   \[\leadsto x + \frac{x}{\frac{z}{y}} \]
10. Add Preprocessing

Alternative 10: 50.4% 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 84.8%

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

   1. associate-*l/ 85.9%

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

   2. *-commutative 85.9%

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

3. Simplified 85.9%

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

5. Taylor expanded in y around 0 47.1%

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

6. Final simplification 47.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 96.3% 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 2024020 
(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))