Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.5% → 95.7%

Time: 5.3s

Alternatives: 7

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: 84.5% 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: 95.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{-98}:\\ \;\;\;\;\frac{x\_m \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 8e-98) (/ (* x_m (+ y z)) z) (* x_m (+ 1.0 (/ 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 <= 8e-98) {
		tmp = (x_m * (y + z)) / z;
	} else {
		tmp = x_m * (1.0 + (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 <= 8d-98) then
        tmp = (x_m * (y + z)) / z
    else
        tmp = x_m * (1.0d0 + (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 <= 8e-98) {
		tmp = (x_m * (y + z)) / z;
	} else {
		tmp = x_m * (1.0 + (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 <= 8e-98:
		tmp = (x_m * (y + z)) / z
	else:
		tmp = x_m * (1.0 + (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 <= 8e-98)
		tmp = Float64(Float64(x_m * Float64(y + z)) / z);
	else
		tmp = Float64(x_m * Float64(1.0 + 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 <= 8e-98)
		tmp = (x_m * (y + z)) / z;
	else
		tmp = x_m * (1.0 + (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, 8e-98], N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999951e-98

   1. Initial program 89.4%

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

   if 7.99999999999999951e-98 < x

   1. Initial program 86.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 99.9%

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

Alternative 2: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y -2.95e-92)
    (/ (* x_m y) z)
    (if (<= y 2.1e-16) x_m (* 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 (y <= -2.95e-92) {
		tmp = (x_m * y) / z;
	} else if (y <= 2.1e-16) {
		tmp = x_m;
	} else {
		tmp = 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 (y <= (-2.95d-92)) then
        tmp = (x_m * y) / z
    else if (y <= 2.1d-16) then
        tmp = x_m
    else
        tmp = 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 (y <= -2.95e-92) {
		tmp = (x_m * y) / z;
	} else if (y <= 2.1e-16) {
		tmp = x_m;
	} else {
		tmp = 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 y <= -2.95e-92:
		tmp = (x_m * y) / z
	elif y <= 2.1e-16:
		tmp = x_m
	else:
		tmp = 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 (y <= -2.95e-92)
		tmp = Float64(Float64(x_m * y) / z);
	elseif (y <= 2.1e-16)
		tmp = x_m;
	else
		tmp = Float64(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 (y <= -2.95e-92)
		tmp = (x_m * y) / z;
	elseif (y <= 2.1e-16)
		tmp = x_m;
	else
		tmp = 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[LessEqual[y, -2.95e-92], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 2.1e-16], x$95$m, N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.95e-92

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x \cdot y\right)}, z\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 73.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right) \]

   5. Simplified 73.6%

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

   if -2.95e-92 < y < 2.1000000000000001e-16

   1. Initial program 87.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 78.1%

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

   if 2.1000000000000001e-16 < y

   1. Initial program 85.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 91.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 91.2%

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

   5. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. rgt-mult-inverse N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z}{y}\right)}\right) \]

      15. /-lowering-/.f64 68.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right) \]

   7. Simplified 68.5%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 75.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   9. Applied egg-rr 75.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-18}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* y (/ x_m z))))
   (* x_s (if (<= y -4.4e-92) t_0 (if (<= y 6.6e-18) x_m t_0)))))

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 * (x_m / z);
	double tmp;
	if (y <= -4.4e-92) {
		tmp = t_0;
	} else if (y <= 6.6e-18) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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 * (x_m / z)
    if (y <= (-4.4d-92)) then
        tmp = t_0
    else if (y <= 6.6d-18) then
        tmp = x_m
    else
        tmp = t_0
    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 * (x_m / z);
	double tmp;
	if (y <= -4.4e-92) {
		tmp = t_0;
	} else if (y <= 6.6e-18) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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 * (x_m / z)
	tmp = 0
	if y <= -4.4e-92:
		tmp = t_0
	elif y <= 6.6e-18:
		tmp = x_m
	else:
		tmp = t_0
	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(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -4.4e-92)
		tmp = t_0;
	elseif (y <= 6.6e-18)
		tmp = x_m;
	else
		tmp = t_0;
	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 * (x_m / z);
	tmp = 0.0;
	if (y <= -4.4e-92)
		tmp = t_0;
	elseif (y <= 6.6e-18)
		tmp = x_m;
	else
		tmp = t_0;
	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[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.4e-92], t$95$0, If[LessEqual[y, 6.6e-18], x$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.39999999999999974e-92 or 6.6000000000000003e-18 < y

   1. Initial program 89.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 92.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 92.2%

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

   5. Taylor expanded in y around inf

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

      1. *-rgt-identity N/A

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

      2. rgt-mult-inverse N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z}{y}\right)}\right) \]

      15. /-lowering-/.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right) \]

   7. Simplified 69.4%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 73.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   9. Applied egg-rr 73.2%

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

   if -4.39999999999999974e-92 < y < 6.6000000000000003e-18

   1. Initial program 87.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 78.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-92}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y z))))
   (* x_s (if (<= y -4.4e-92) t_0 (if (<= y 2.9e-16) x_m t_0)))))

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);
	double tmp;
	if (y <= -4.4e-92) {
		tmp = t_0;
	} else if (y <= 2.9e-16) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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)
    if (y <= (-4.4d-92)) then
        tmp = t_0
    else if (y <= 2.9d-16) then
        tmp = x_m
    else
        tmp = t_0
    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);
	double tmp;
	if (y <= -4.4e-92) {
		tmp = t_0;
	} else if (y <= 2.9e-16) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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)
	tmp = 0
	if y <= -4.4e-92:
		tmp = t_0
	elif y <= 2.9e-16:
		tmp = x_m
	else:
		tmp = t_0
	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(x_m * Float64(y / z))
	tmp = 0.0
	if (y <= -4.4e-92)
		tmp = t_0;
	elseif (y <= 2.9e-16)
		tmp = x_m;
	else
		tmp = t_0;
	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);
	tmp = 0.0;
	if (y <= -4.4e-92)
		tmp = t_0;
	elseif (y <= 2.9e-16)
		tmp = x_m;
	else
		tmp = t_0;
	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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.4e-92], t$95$0, If[LessEqual[y, 2.9e-16], x$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.39999999999999974e-92 or 2.8999999999999998e-16 < y

   1. Initial program 89.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 92.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 92.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{z}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 68.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 68.8%

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

   if -4.39999999999999974e-92 < y < 2.8999999999999998e-16

   1. Initial program 87.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

      18. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

      23. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

      24. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

      25. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

      26. *-rgt-identity 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 78.1%

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

Alternative 5: 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 #s(literal 1 binary64) 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 88.3%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

   9. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

   11. lft-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

   12. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

   17. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

   18. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

   19. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

   21. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

   22. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

   23. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

   24. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

   25. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

   26. *-rgt-identity 95.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

3. Simplified 95.1%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{y}{z} \cdot x\right), \color{blue}{x}\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{\frac{z}{y}} \cdot x\right), x\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1 \cdot x}{\frac{z}{y}}\right), x\right) \]

   7. *-lft-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{\frac{z}{y}}\right), x\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{z}{y}\right)\right), x\right) \]

   9. /-lowering-/.f64 95.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, y\right)\right), x\right) \]

6. Applied egg-rr 95.5%

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

7. Final simplification 95.5%

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

Alternative 6: 96.0% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* x_m (+ 1.0 (/ 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 * (1.0 + (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 * (1.0d0 + (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 * (1.0 + (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 * (1.0 + (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(1.0 + 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 * (1.0 + (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[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

   9. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

   11. lft-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

   12. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

   17. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

   18. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

   19. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

   21. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

   22. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

   23. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

   24. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

   25. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

   26. *-rgt-identity 95.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

3. Simplified 95.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{y}{z}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 51.3% 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 #s(literal 1 binary64) 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 88.3%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y + z}{z}\right)}\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(y + z\right)}{z}\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot \left(y + z\right)}{z}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot \color{blue}{\left(y + z\right)}\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z}}{z} \cdot y + \color{blue}{\frac{\frac{z}{z}}{z} \cdot z}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{z}{z} \cdot y}{z} + \color{blue}{\frac{\frac{z}{z}}{z}} \cdot z\right)\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot y}{z} + \frac{\frac{\color{blue}{z}}{z}}{z} \cdot z\right)\right) \]

   9. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{\color{blue}{\frac{z}{z}}}{z} \cdot z\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{1}{z} \cdot z\right)\right) \]

   11. lft-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + 1\right)\right) \]

   12. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{z} + \frac{z}{\color{blue}{z}}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z}{z} + \color{blue}{\frac{y}{z}}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{z}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\color{blue}{y}}{z}\right)\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot 1}{z}\right)\right)\right) \]

   17. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{y \cdot \frac{z}{z}}{z}\right)\right)\right) \]

   18. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y \cdot z}{z}}{z}\right)\right)\right) \]

   19. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{y}{z} \cdot z}{z}\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{z \cdot \frac{y}{z}}{z}\right)\right)\right) \]

   21. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(z \cdot \frac{y}{z}\right), \color{blue}{z}\right)\right)\right) \]

   22. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y}{z} \cdot z\right), z\right)\right)\right) \]

   23. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y \cdot z}{z}\right), z\right)\right)\right) \]

   24. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \frac{z}{z}\right), z\right)\right)\right) \]

   25. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 1\right), z\right)\right)\right) \]

   26. *-rgt-identity 95.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, z\right)\right)\right) \]

3. Simplified 95.1%

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

5. Taylor expanded in y around 0

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

7. Simplified 45.0%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 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 2024160 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ z (+ y z))))

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