Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.8%

Time: 8.6s

Alternatives: 21

Speedup: 2.6×

• 64.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+279}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    x_s
    (* y_s (if (<= t_0 1e+279) (/ t_0 z) (/ (* y_m (/ (cosh x_m) z)) x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 1e+279) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 1d+279) then
        tmp = t_0 / z
    else
        tmp = (y_m * (cosh(x_m) / z)) / x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 1e+279) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 1e+279:
		tmp = t_0 / z
	else:
		tmp = (y_m * (math.cosh(x_m) / z)) / x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 1e+279)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 1e+279)
		tmp = t_0 / z;
	else
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 1e+279], N[(t$95$0 / z), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000006e279

   1. Initial program 97.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   if 1.00000000000000006e279 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 57.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 4 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z} \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 4e+237)
     (* (/ (/ y_m x_m) z) (fma 0.5 (* x_m x_m) 1.0))
     (/ (* y_m (/ (cosh x_m) z)) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 4e+237) {
		tmp = ((y_m / x_m) / z) * fma(0.5, (x_m * x_m), 1.0);
	} else {
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 4e+237)
		tmp = Float64(Float64(Float64(y_m / x_m) / z) * fma(0.5, Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 4e+237], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 3.99999999999999976e237

   1. Initial program 94.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 79.5

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 79.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      6. lower-/.f64 75.8

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z}} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

   7. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]

   if 3.99999999999999976e237 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 64.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+148}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z} \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+148)
     (* (/ (/ y_m x_m) z) (fma 0.5 (* x_m x_m) 1.0))
     (/
      (/
       (*
        y_m
        (fma
         (fma
          (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
          (* x_m x_m)
          0.5)
         (* x_m x_m)
         1.0))
       z)
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+148) {
		tmp = ((y_m / x_m) / z) * fma(0.5, (x_m * x_m), 1.0);
	} else {
		tmp = ((y_m * fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+148)
		tmp = Float64(Float64(Float64(y_m / x_m) / z) * fma(0.5, Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(y_m * fma(fma(fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+148], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e148

   1. Initial program 94.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 78.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 78.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      6. lower-/.f64 74.8

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z}} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

   7. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]

   if 1e148 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 66.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 60.6

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 60.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. lower-*.f64 89.8

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{x}}{z} \]

   7. Applied rewrites 89.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot y}{x}}{z} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot y}{x}}{z} \]

      14. lower-*.f64 94.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot y}{x}}{z} \]

   10. Applied rewrites 94.5%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot y}{x}}{z} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}{z}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

       3. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

       5. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]

       6. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]

   12. Applied rewrites 97.2%

       \[\leadsto \color{blue}{\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+148}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z} \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+148)
     (* (/ (/ y_m x_m) z) (fma 0.5 (* x_m x_m) 1.0))
     (/
      (/
       (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y_m)
       z)
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+148) {
		tmp = ((y_m / x_m) / z) * fma(0.5, (x_m * x_m), 1.0);
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+148)
		tmp = Float64(Float64(Float64(y_m / x_m) / z) * fma(0.5, Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+148], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e148

   1. Initial program 94.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 78.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 78.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      6. lower-/.f64 74.8

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z}} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

   7. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]

   if 1e148 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 66.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 60.6

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 60.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}}{x} \]

      10. lower-*.f64 92.5

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{z}}{x} \]

   7. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 4 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z} \cdot \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 4e+237)
     (* (/ (/ y_m x_m) z) (fma 0.5 (* x_m x_m) 1.0))
     (/
      (* y_m (/ (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0) z))
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 4e+237) {
		tmp = ((y_m / x_m) / z) * fma(0.5, (x_m * x_m), 1.0);
	} else {
		tmp = (y_m * (fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 4e+237)
		tmp = Float64(Float64(Float64(y_m / x_m) / z) * fma(0.5, Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 4e+237], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 3.99999999999999976e237

   1. Initial program 94.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 79.5

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 79.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right)} \]

      6. lower-/.f64 75.8

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z}} \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

   7. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]

   if 3.99999999999999976e237 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 64.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1}{z}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}}{z}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)}{z}}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)}{z}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{z}}{x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)}{z}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)}{z}}{x} \]

      9. lower-*.f64 92.0

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)}{z}}{x} \]

   7. Applied rewrites 92.0%

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}}{z}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right)}{z}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 92.0%

       \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \color{blue}{x} \cdot x, 1\right)}{z}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e+296)
     (/ (/ y_m x_m) z)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e+296) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e+296)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+296], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999996e296

   1. Initial program 94.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if 1.99999999999999996e296 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 62.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 46.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 46.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 10^{+148}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 1e+148)
     (/ (/ y_m x_m) z)
     (/ (/ y_m z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+148) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z) <= 1d+148) then
        tmp = (y_m / x_m) / z
    else
        tmp = (y_m / z) / x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z) <= 1e+148) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_m / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z) <= 1e+148:
		tmp = (y_m / x_m) / z
	else:
		tmp = (y_m / z) / x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 1e+148)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(y_m / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 1e+148)
		tmp = (y_m / x_m) / z;
	else
		tmp = (y_m / z) / x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+148], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e148

   1. Initial program 94.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if 1e148 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 66.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 34.4

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

   7. Applied rewrites 34.4%

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

Alternative 8: 52.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z} \leq 2 \cdot 10^{-80}:\\ \;\;\;\;\frac{1 \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* (cosh x_m) (/ y_m x_m)) z) 2e-80)
     (/ (* 1.0 y_m) (* z x_m))
     (/ (/ y_m z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-80) {
		tmp = (1.0 * y_m) / (z * x_m);
	} else {
		tmp = (y_m / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x_m) * (y_m / x_m)) / z) <= 2d-80) then
        tmp = (1.0d0 * y_m) / (z * x_m)
    else
        tmp = (y_m / z) / x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x_m) * (y_m / x_m)) / z) <= 2e-80) {
		tmp = (1.0 * y_m) / (z * x_m);
	} else {
		tmp = (y_m / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((math.cosh(x_m) * (y_m / x_m)) / z) <= 2e-80:
		tmp = (1.0 * y_m) / (z * x_m)
	else:
		tmp = (y_m / z) / x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z) <= 2e-80)
		tmp = Float64(Float64(1.0 * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(y_m / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((cosh(x_m) * (y_m / x_m)) / z) <= 2e-80)
		tmp = (1.0 * y_m) / (z * x_m);
	else
		tmp = (y_m / z) / x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-80], N[(N[(1.0 * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999992e-80

   1. Initial program 94.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 86.5

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 86.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg 83.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1} \cdot y}{z \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.0%

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

   if 1.99999999999999992e-80 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 69.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 40.4

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

   7. Applied rewrites 40.4%

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

Alternative 9: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{+50}:\\ \;\;\;\;y\_m \cdot \frac{\cosh x\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right) \cdot x\_m\right) \cdot x\_m, x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 3.6e+50)
     (* y_m (/ (cosh x_m) (* z x_m)))
     (/
      (/
       (*
        (fma
         (*
          (* (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664) x_m)
          x_m)
         (* x_m x_m)
         1.0)
        y_m)
       x_m)
      z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 3.6e+50) {
		tmp = y_m * (cosh(x_m) / (z * x_m));
	} else {
		tmp = ((fma(((fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664) * x_m) * x_m), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 3.6e+50)
		tmp = Float64(y_m * Float64(cosh(x_m) / Float64(z * x_m)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664) * x_m) * x_m), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 3.6e+50], N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.59999999999999986e50

   1. Initial program 85.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.1

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

   4. Applied rewrites 88.1%

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

   if 3.59999999999999986e50 < x

   1. Initial program 72.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 70.4

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. lower-*.f64 98.2

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{x}}{z} \]

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot y}{x}}{z} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot y}{x}}{z} \]

      14. lower-*.f64 100.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot y}{x}}{z} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot y}{x}}{z} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]
   12. Step-by-step derivation

   13. Applied rewrites 100.0%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right) \cdot x\_m\right) \cdot x\_m, x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 2.2e-111)
     (/
      (/
       (*
        (fma
         (*
          (* (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664) x_m)
          x_m)
         (* x_m x_m)
         1.0)
        y_m)
       x_m)
      z)
     (/
      (/
       (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y_m)
       z)
      x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.2e-111) {
		tmp = ((fma(((fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664) * x_m) * x_m), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = ((fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2.2e-111)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664) * x_m) * x_m), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 2.2e-111], N[(N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.2e-111

   1. Initial program 75.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 70.4

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. lower-*.f64 88.2

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{x}}{z} \]

   7. Applied rewrites 88.2%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot y}{x}}{z} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot y}{x}}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot y}{x}}{z} \]

      14. lower-*.f64 90.6

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot y}{x}}{z} \]

   10. Applied rewrites 90.6%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot y}{x}}{z} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]
   12. Step-by-step derivation

   13. Applied rewrites 90.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]

   if 2.2e-111 < y

   1. Initial program 95.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 87.7

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 87.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{z}}}{x} \]

      10. lower-*.f64 92.8

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{z}}{x} \]

   7. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.9 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 5.9e+113)
     (/
      (/
       (* y_m (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
       x_m)
      z)
     (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 5.9e+113) {
		tmp = ((y_m * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / x_m) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 5.9e+113)
		tmp = Float64(Float64(Float64(y_m * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 5.9e+113], N[(N[(N[(y$95$m * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.90000000000000023e113

   1. Initial program 80.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]

   5. Applied rewrites 88.3%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}}{z} \]

   if 5.90000000000000023e113 < y

   1. Initial program 94.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 79.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 79.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}{x}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}}{x} \]

      10. lower-*.f64 96.2

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}}{z}}{x} \]

   7. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 90.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.9 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 5.9e+113)
     (/
      (/
       (* (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0) y_m)
       x_m)
      z)
     (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 5.9e+113) {
		tmp = ((fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0) * y_m) / x_m) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 5.9e+113)
		tmp = Float64(Float64(Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) * y_m) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 5.9e+113], N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.90000000000000023e113

   1. Initial program 80.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 73.3

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 73.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. lower-*.f64 88.3

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}}{x}}{z} \]

   7. Applied rewrites 88.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 88.2%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \color{blue}{x} \cdot x, 1\right) \cdot y}{x}}{z} \]

   if 5.90000000000000023e113 < y

   1. Initial program 94.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 79.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 79.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}{x}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}}{x} \]

      10. lower-*.f64 96.2

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}}{z}}{x} \]

   7. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 85.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{0.5 \cdot \left(x\_m \cdot x\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 5.3e+150)
     (/
      (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (/ (* y_m (/ (* 0.5 (* x_m x_m)) z)) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 5.3e+150) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = (y_m * ((0.5 * (x_m * x_m)) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 5.3e+150)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(0.5 * Float64(x_m * x_m)) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 5.3e+150], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.30000000000000013e150

   1. Initial program 86.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 78.8

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 78.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg 80.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   if 5.30000000000000013e150 < x

   1. Initial program 65.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 65.0

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 65.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}}{x}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{0.5 \cdot \left(x \cdot x\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 85.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{0.5 \cdot \left(x\_m \cdot x\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 5.3e+150)
     (/
      (* (fma (* 0.041666666666666664 (* x_m x_m)) (* x_m x_m) 1.0) y_m)
      (* z x_m))
     (/ (* y_m (/ (* 0.5 (* x_m x_m)) z)) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 5.3e+150) {
		tmp = (fma((0.041666666666666664 * (x_m * x_m)), (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = (y_m * ((0.5 * (x_m * x_m)) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 5.3e+150)
		tmp = Float64(Float64(fma(Float64(0.041666666666666664 * Float64(x_m * x_m)), Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(0.5 * Float64(x_m * x_m)) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 5.3e+150], N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.30000000000000013e150

   1. Initial program 86.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 78.8

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 78.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg 80.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right) \cdot y}{z \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.2%

       \[\leadsto \frac{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \color{blue}{x} \cdot x, 1\right) \cdot y}{z \cdot x} \]

   if 5.30000000000000013e150 < x

   1. Initial program 65.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 65.0

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 65.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}}{x}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{0.5 \cdot \left(x \cdot x\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 82.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-150}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 1.5e-150)
     (* y_m (/ (/ (fma (* x_m x_m) 0.5 1.0) z) x_m))
     (/ (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e-150) {
		tmp = y_m * ((fma((x_m * x_m), 0.5, 1.0) / z) / x_m);
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 1.0) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.5e-150)
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / z) / x_m));
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 1.5e-150], N[(y$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.5000000000000001e-150

   1. Initial program 75.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{2}}{z}} + \frac{1}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{2}}}{z} + \frac{1}{z}}{x} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\color{blue}{{x}^{2} \cdot \frac{\frac{1}{2}}{z}} + \frac{1}{z}}{x} \]

      4. metadata-eval N/A

         \[\leadsto y \cdot \frac{{x}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{z} + \frac{1}{z}}{x} \]

      5. associate-*r/ N/A

         \[\leadsto y \cdot \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{z}\right)} + \frac{1}{z}}{x} \]

      6. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}} \]

   7. Applied rewrites 83.7%

      \[\leadsto y \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}}{x}} \]

   if 1.5000000000000001e-150 < y

   1. Initial program 94.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 74.2

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 74.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}{x}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{z}}}{x} \]

      10. lower-*.f64 88.4

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot y}}{z}}{x} \]

   7. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 81.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0.5 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot y\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 8e+21)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))
     (/ (/ (* (* 0.5 (* x_m x_m)) y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 8e+21) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = (((0.5 * (x_m * x_m)) * y_m) / z) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 8e+21)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(x_m * x_m)) * y_m) / z) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 8e+21], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8e21

   1. Initial program 85.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 72.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   if 8e21 < x

   1. Initial program 75.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 53.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 53.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.3%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{z}}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{z}}{x}} \]

   10. Applied rewrites 85.7%

       \[\leadsto \color{blue}{\frac{\frac{\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot y}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 80.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{0.5 \cdot \left(x\_m \cdot x\_m\right)}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.9e+73)
     (/ (* (fma 0.5 (* x_m x_m) 1.0) y_m) (* z x_m))
     (/ (* y_m (/ (* 0.5 (* x_m x_m)) z)) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.9e+73) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y_m) / (z * x_m);
	} else {
		tmp = (y_m * ((0.5 * (x_m * x_m)) / z)) / x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.9e+73)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(0.5 * Float64(x_m * x_m)) / z)) / x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.9e+73], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.90000000000000011e73

   1. Initial program 85.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 71.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 71.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right)\right)\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   if 1.90000000000000011e73 < x

   1. Initial program 72.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 56.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{z}}{x}} \]

   10. Applied rewrites 87.5%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{0.5 \cdot \left(x \cdot x\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 68.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{1 \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot y\_m}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.4)
     (/ (* 1.0 y_m) (* z x_m))
     (/ (* (* 0.5 (* x_m x_m)) y_m) (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (1.0 * y_m) / (z * x_m);
	} else {
		tmp = ((0.5 * (x_m * x_m)) * y_m) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (1.0d0 * y_m) / (z * x_m)
    else
        tmp = ((0.5d0 * (x_m * x_m)) * y_m) / (z * x_m)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (1.0 * y_m) / (z * x_m);
	} else {
		tmp = ((0.5 * (x_m * x_m)) * y_m) / (z * x_m);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.4:
		tmp = (1.0 * y_m) / (z * x_m)
	else:
		tmp = ((0.5 * (x_m * x_m)) * y_m) / (z * x_m)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(Float64(1.0 * y_m) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(x_m * x_m)) * y_m) / Float64(z * x_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = (1.0 * y_m) / (z * x_m);
	else
		tmp = ((0.5 * (x_m * x_m)) * y_m) / (z * x_m);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(1.0 * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 80.2

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 80.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg 83.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1} \cdot y}{z \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.2%

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

   if 1.3999999999999999 < x

   1. Initial program 76.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 51.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.1%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

   10. Applied rewrites 53.6%

       \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot y}{z \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 67.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4:\\ \;\;\;\;\frac{1 \cdot y\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{0.5 \cdot \left(x\_m \cdot x\_m\right)}{z \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.4)
     (/ (* 1.0 y_m) (* z x_m))
     (* y_m (/ (* 0.5 (* x_m x_m)) (* z x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (1.0 * y_m) / (z * x_m);
	} else {
		tmp = y_m * ((0.5 * (x_m * x_m)) / (z * x_m));
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = (1.0d0 * y_m) / (z * x_m)
    else
        tmp = y_m * ((0.5d0 * (x_m * x_m)) / (z * x_m))
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = (1.0 * y_m) / (z * x_m);
	} else {
		tmp = y_m * ((0.5 * (x_m * x_m)) / (z * x_m));
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.4:
		tmp = (1.0 * y_m) / (z * x_m)
	else:
		tmp = y_m * ((0.5 * (x_m * x_m)) / (z * x_m))
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(Float64(1.0 * y_m) / Float64(z * x_m));
	else
		tmp = Float64(y_m * Float64(Float64(0.5 * Float64(x_m * x_m)) / Float64(z * x_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = (1.0 * y_m) / (z * x_m);
	else
		tmp = y_m * ((0.5 * (x_m * x_m)) / (z * x_m));
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.4], N[(N[(1.0 * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. lower-*.f64 80.2

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 80.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      11. remove-double-neg 83.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1} \cdot y}{z \cdot x} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.2%

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

   if 1.3999999999999999 < x

   1. Initial program 76.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. lower-*.f64 51.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \frac{y}{x}}{z} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.1%

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot \frac{y}{x}}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}}{z} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{x}}}{z} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{x \cdot z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{z \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{z \cdot x}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \color{blue}{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

   10. Applied rewrites 44.9%

       \[\leadsto \color{blue}{y \cdot \frac{0.5 \cdot \left(x \cdot x\right)}{z \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 48.9% accurate, 5.8× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (* 1.0 y_m) (* z x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((1.0 * y_m) / (z * x_m)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((1.0d0 * y_m) / (z * x_m)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((1.0 * y_m) / (z * x_m)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * ((1.0 * y_m) / (z * x_m)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(1.0 * y_m) / Float64(z * x_m))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * ((1.0 * y_m) / (z * x_m)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(1.0 * y$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{y}{x}}{z} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

   9. lower-*.f64 76.7

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

5. Applied rewrites 76.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{y}{x}}{z} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}{z}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{y}{x}}}{z} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{\frac{y}{x}}}{z} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x}}}{z} \]

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{x \cdot z}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

   8. remove-double-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot y}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right)\right)} \]

   11. remove-double-neg 77.2

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{\color{blue}{z \cdot x}} \]

7. Applied rewrites 77.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{z \cdot x}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1} \cdot y}{z \cdot x} \]
9. Step-by-step derivation

10. Applied rewrites 50.3%

    \[\leadsto \frac{\color{blue}{1} \cdot y}{z \cdot x} \]
11. Add Preprocessing

Alternative 21: 48.5% accurate, 5.8× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* y_m (/ 1.0 (* z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m * (1.0 / (z * x_m))));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (y_m * (1.0d0 / (z * x_m))))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (y_m * (1.0 / (z * x_m))));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (y_m * (1.0 / (z * x_m))))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(y_m * Float64(1.0 / Float64(z * x_m)))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (y_m * (1.0 / (z * x_m))));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(y$95$m * N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 83.2

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

4. Applied rewrites 83.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 50.0%

   \[\leadsto y \cdot \frac{\color{blue}{1}}{z \cdot x} \]
8. Add Preprocessing

Developer Target 1: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

  (/ (* (cosh x) (/ y x)) z))