Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 95.1%

Time: 9.4s

Alternatives: 17

Speedup: 2.6×

• Could not converge on a ground truth

  1. x = 2.892578603931132e+261
  2. y = 4.1943634620738593e-249
  3. z = 1.0749307193865454e+61

• 64.2% 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: 84.8% 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: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{x\_m}}{z}\\ \mathbf{elif}\;x\_m \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\cosh x\_m}{z \cdot x\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.45e-148)
    (/ (/ y x_m) z)
    (if (<= x_m 2.6e+77)
      (* (/ (cosh x_m) (* z x_m)) y)
      (*
       (/
        (/ (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0) z)
        x_m)
       y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.45e-148) {
		tmp = (y / x_m) / z;
	} else if (x_m <= 2.6e+77) {
		tmp = (cosh(x_m) / (z * x_m)) * y;
	} else {
		tmp = ((fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.45e-148)
		tmp = Float64(Float64(y / x_m) / z);
	elseif (x_m <= 2.6e+77)
		tmp = Float64(Float64(cosh(x_m) / Float64(z * x_m)) * y);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.45e-148], N[(N[(y / x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x$95$m, 2.6e+77], N[(N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.4499999999999999e-148

   1. Initial program 84.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 61.9

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

   5. Applied rewrites 61.9%

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

   if 1.4499999999999999e-148 < x < 2.6000000000000002e77

   1. Initial program 90.9%

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

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if 2.6000000000000002e77 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\cosh x}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{y}{x\_m} \cdot \cosh x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (/ y x_m) (cosh x_m))))
   (*
    x_s
    (if (<= t_0 INFINITY)
      (/ t_0 z)
      (*
       (/
        (/ (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0) z)
        x_m)
       y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (y / x_m) * cosh(x_m);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = ((fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(y / x_m) * cosh(x_m))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 96.1%

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

   if +inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq \infty:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y}{x\_m} \cdot \cosh x\_m \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y + \frac{y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ y x_m) (cosh x_m)) 2e+290)
    (/ (+ (* (* 0.5 x_m) y) (/ y x_m)) z)
    (*
     (/
      (/ (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0) z)
      x_m)
     y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((y / x_m) * cosh(x_m)) <= 2e+290) {
		tmp = (((0.5 * x_m) * y) + (y / x_m)) / z;
	} else {
		tmp = ((fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(y / x_m) * cosh(x_m)) <= 2e+290)
		tmp = Float64(Float64(Float64(Float64(0.5 * x_m) * y) + Float64(y / x_m)) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision], 2e+290], N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y), $MachinePrecision] + N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000012e290

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 75.7%

      \[\leadsto \frac{\frac{y}{x} + \color{blue}{\left(0.5 \cdot x\right) \cdot y}}{z} \]

   if 2.00000000000000012e290 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 60.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)}{z}}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x\_m} \cdot \cosh x\_m}{z} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\_m\right) \cdot y + \frac{y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot \frac{y}{z}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (/ y x_m) (cosh x_m)) z) 2e+52)
    (/ (+ (* (* 0.5 x_m) y) (/ y x_m)) z)
    (/ (* (fma (* x_m x_m) 0.5 1.0) (/ y z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((y / x_m) * cosh(x_m)) / z) <= 2e+52) {
		tmp = (((0.5 * x_m) * y) + (y / x_m)) / z;
	} else {
		tmp = (fma((x_m * x_m), 0.5, 1.0) * (y / z)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y / x_m) * cosh(x_m)) / z) <= 2e+52)
		tmp = Float64(Float64(Float64(Float64(0.5 * x_m) * y) + Float64(y / x_m)) / z);
	else
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * Float64(y / z)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(y / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+52], N[(N[(N[(N[(0.5 * x$95$m), $MachinePrecision] * y), $MachinePrecision] + N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2e52

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.9%

      \[\leadsto \frac{\frac{y}{x} + \color{blue}{\left(0.5 \cdot x\right) \cdot y}}{z} \]

   if 2e52 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 73.2

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

   8. Applied rewrites 73.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 72.3%

      \[\leadsto \frac{\frac{y}{x} + \color{blue}{\left(0.5 \cdot x\right) \cdot y}}{z} \]

   if +inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-/l/ N/A

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

      8. distribute-lft1-in N/A

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

      9. associate-*r/ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 6.5%

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

   10. Applied rewrites 31.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq \infty:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if +inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-/l/ N/A

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

      8. distribute-lft1-in N/A

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

      9. associate-*r/ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 6.5%

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

   10. Applied rewrites 31.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{x} \cdot \cosh x \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1.35e-32)
    (/
     (*
      (/
       (fma
        (*
         (fma
          (* (fma 0.001388888888888889 (* x_m x_m) 0.041666666666666664) x_m)
          x_m
          0.5)
         x_m)
        x_m
        1.0)
       x_m)
      y)
     z)
    (*
     (/ y z)
     (/
      (fma
       (fma
        (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
        (* x_m x_m)
        0.5)
       (* x_m x_m)
       1.0)
      x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.35e-32) {
		tmp = ((fma((fma((fma(0.001388888888888889, (x_m * x_m), 0.041666666666666664) * x_m), x_m, 0.5) * x_m), x_m, 1.0) / x_m) * y) / z;
	} else {
		tmp = (y / z) * (fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.35e-32)
		tmp = Float64(Float64(Float64(fma(Float64(fma(Float64(fma(0.001388888888888889, Float64(x_m * x_m), 0.041666666666666664) * x_m), x_m, 0.5) * x_m), x_m, 1.0) / x_m) * y) / z);
	else
		tmp = Float64(Float64(y / z) * Float64(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) / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.35e-32], N[(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 + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(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] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.3499999999999999e-32

   1. Initial program 80.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 89.9%

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

   9. Applied rewrites 89.9%

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

   if 1.3499999999999999e-32 < y

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 91.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x, x, 0.5\right) \cdot x, x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1.35e-32)
    (/
     (*
      (/
       (fma
        (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
        (* x_m x_m)
        1.0)
       x_m)
      y)
     z)
    (*
     (/ y z)
     (/
      (fma
       (fma
        (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
        (* x_m x_m)
        0.5)
       (* x_m x_m)
       1.0)
      x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.35e-32) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y) / z;
	} else {
		tmp = (y / z) * (fma(fma(fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.35e-32)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y) / z);
	else
		tmp = Float64(Float64(y / z) * Float64(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) / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.35e-32], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(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] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.3499999999999999e-32

   1. Initial program 80.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.9%

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

   if 1.3499999999999999e-32 < y

   1. Initial program 93.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 86.5

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

   5. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 91.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.45 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 4.45e+151)
    (/
     (*
      (/
       (fma
        (fma (* 0.001388888888888889 (* x_m x_m)) (* x_m x_m) 0.5)
        (* x_m x_m)
        1.0)
       x_m)
      y)
     z)
    (*
     (/
      (/ (fma (* (fma 0.041666666666666664 (* x_m x_m) 0.5) x_m) x_m 1.0) z)
      x_m)
     y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 4.45e+151) {
		tmp = ((fma(fma((0.001388888888888889 * (x_m * x_m)), (x_m * x_m), 0.5), (x_m * x_m), 1.0) / x_m) * y) / z;
	} else {
		tmp = ((fma((fma(0.041666666666666664, (x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 4.45e+151)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.001388888888888889 * Float64(x_m * x_m)), Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / x_m) * y) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(0.041666666666666664, Float64(x_m * x_m), 0.5) * x_m), x_m, 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 4.45e+151], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.4500000000000003e151

   1. Initial program 83.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.4%

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

   if 4.4500000000000003e151 < y

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 95.8%

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

Alternative 10: 84.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{x\_m}}{z}\\ \mathbf{elif}\;x\_m \leq 2.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{z \cdot x\_m} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot y}{x\_m}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.45e-148)
    (/ (/ y x_m) z)
    (if (<= x_m 2.2e+141)
      (*
       (/
        (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0)
        (* z x_m))
       y)
      (/ (/ (* (fma (* x_m x_m) 0.5 1.0) y) x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.45e-148) {
		tmp = (y / x_m) / z;
	} else if (x_m <= 2.2e+141) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) / (z * x_m)) * y;
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) * y) / x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.45e-148)
		tmp = Float64(Float64(y / x_m) / z);
	elseif (x_m <= 2.2e+141)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) / Float64(z * x_m)) * y);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * y) / x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.45e-148], N[(N[(y / x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x$95$m, 2.2e+141], 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] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.4499999999999999e-148

   1. Initial program 84.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 61.9

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

   5. Applied rewrites 61.9%

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

   if 1.4499999999999999e-148 < x < 2.2e141

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 77.4%

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

   if 2.2e141 < x

   1. Initial program 83.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.8%

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

Alternative 11: 88.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, x\_m \cdot x\_m, 1\right) \cdot y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0 \cdot x\_m, x\_m, 1\right)}{z}}{x\_m} \cdot y\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x_m x_m) 0.5)))
   (*
    x_s
    (if (<= y 2.9e+149)
      (/ (/ (* (fma t_0 (* x_m x_m) 1.0) y) x_m) z)
      (* (/ (/ (fma (* t_0 x_m) x_m 1.0) z) x_m) y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma(0.041666666666666664, (x_m * x_m), 0.5);
	double tmp;
	if (y <= 2.9e+149) {
		tmp = ((fma(t_0, (x_m * x_m), 1.0) * y) / x_m) / z;
	} else {
		tmp = ((fma((t_0 * x_m), x_m, 1.0) / z) / x_m) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = fma(0.041666666666666664, Float64(x_m * x_m), 0.5)
	tmp = 0.0
	if (y <= 2.9e+149)
		tmp = Float64(Float64(Float64(fma(t_0, Float64(x_m * x_m), 1.0) * y) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(t_0 * x_m), x_m, 1.0) / z) / x_m) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, 2.9e+149], N[(N[(N[(N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9000000000000002e149

   1. Initial program 83.7%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 87.9%

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

   7. Applied rewrites 89.8%

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

   if 2.9000000000000002e149 < y

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

Alternative 12: 85.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right) \cdot y}{x\_m}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.2e+141)
    (/
     (* (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0) y)
     (* z x_m))
    (/ (/ (* (fma (* x_m x_m) 0.5 1.0) y) x_m) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.2e+141) {
		tmp = (fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0) * y) / (z * x_m);
	} else {
		tmp = ((fma((x_m * x_m), 0.5, 1.0) * y) / x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.2e+141)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) * y) / x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.2e+141], 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), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e141

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 79.8%

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

   if 2.2e141 < x

   1. Initial program 83.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.8%

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

Alternative 13: 80.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{+73}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{y}{z}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) 0.5 1.0)))
   (* x_s (if (<= y 1e+73) (/ (/ (* t_0 y) x_m) z) (/ (* t_0 (/ y z)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma((x_m * x_m), 0.5, 1.0);
	double tmp;
	if (y <= 1e+73) {
		tmp = ((t_0 * y) / x_m) / z;
	} else {
		tmp = (t_0 * (y / z)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = fma(Float64(x_m * x_m), 0.5, 1.0)
	tmp = 0.0
	if (y <= 1e+73)
		tmp = Float64(Float64(Float64(t_0 * y) / x_m) / z);
	else
		tmp = Float64(Float64(t_0 * Float64(y / z)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999983e72

   1. Initial program 82.7%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 89.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 76.4%

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

   if 9.99999999999999983e72 < y

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 93.5

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

   8. Applied rewrites 93.5%

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

Alternative 14: 80.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.5, 1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{x\_m} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{y}{z}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma (* x_m x_m) 0.5 1.0)))
   (*
    x_s
    (if (<= y 2.5e-31) (/ (* (/ t_0 x_m) y) z) (/ (* t_0 (/ y z)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma((x_m * x_m), 0.5, 1.0);
	double tmp;
	if (y <= 2.5e-31) {
		tmp = ((t_0 / x_m) * y) / z;
	} else {
		tmp = (t_0 * (y / z)) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = fma(Float64(x_m * x_m), 0.5, 1.0)
	tmp = 0.0
	if (y <= 2.5e-31)
		tmp = Float64(Float64(Float64(t_0 / x_m) * y) / z);
	else
		tmp = Float64(Float64(t_0 * Float64(y / z)) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, 2.5e-31], N[(N[(N[(t$95$0 / x$95$m), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$0 * N[(y / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5e-31

   1. Initial program 80.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.1%

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

   if 2.5e-31 < y

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

Alternative 15: 65.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

   if 1.3999999999999999 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-/l/ N/A

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

      8. distribute-lft1-in N/A

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

      9. associate-*r/ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.8%

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

   10. Applied rewrites 46.2%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if 1.3999999999999999 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. associate-/l/ N/A

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

      8. distribute-lft1-in N/A

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

      9. associate-*r/ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.8%

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

   10. Applied rewrites 46.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.1% accurate, 7.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ y (* z x_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 51.7

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

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{y}{z \cdot x}} \]
6. Add Preprocessing

Developer Target 1: 96.8% 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 2024327 
(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))