Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.3% → 95.5%

Time: 8.1s

Alternatives: 17

Speedup: 2.3×

• 63.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 85.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{y \cdot \cosh x\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \end{array} \end{array} \]

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 1.86e-56)
    (/ (/ y z) x_m)
    (if (<= x_m 3.5e+45)
      (/ (* y (cosh x_m)) (* z x_m))
      (/
       (*
        (/
         (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)
        y)
       z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.86e-56) {
		tmp = (y / z) / x_m;
	} else if (x_m <= 3.5e+45) {
		tmp = (y * cosh(x_m)) / (z * x_m);
	} else {
		tmp = ((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) * y) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.86e-56)
		tmp = Float64(Float64(y / z) / x_m);
	elseif (x_m <= 3.5e+45)
		tmp = Float64(Float64(y * cosh(x_m)) / Float64(z * x_m));
	else
		tmp = Float64(Float64(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) * y) / 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, 1.86e-56], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 3.5e+45], N[(N[(y * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(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] * y), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.85999999999999997e-56

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 1.85999999999999997e-56 < x < 3.50000000000000023e45

   1. Initial program 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 3.50000000000000023e45 < x

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y}{x\_m}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \end{array} \end{array} \end{array} \]

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 (/ (* (cosh x_m) (/ y x_m)) z)))
   (*
    x_s
    (if (<= t_0 INFINITY)
      t_0
      (/
       (*
        (/
         (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)
        y)
       z)))))

C

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y / x_m)) / z)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(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) * y) / 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_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(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] * y), $MachinePrecision] / z), $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 96.3%

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

   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

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 3: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y}{x\_m}}{z} \leq 10^{+76}:\\ \;\;\;\;\frac{y}{z \cdot x\_m} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{z}}{x\_m}\\ \end{array} \end{array} \]

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 (<= (/ (* (cosh x_m) (/ y x_m)) z) 1e+76)
    (*
     (/ y (* z x_m))
     (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0))
    (/ (/ (* (fma 0.5 (* x_m x_m) 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 (((cosh(x_m) * (y / x_m)) / z) <= 1e+76) {
		tmp = (y / (z * x_m)) * fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0);
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 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(cosh(x_m) * Float64(y / x_m)) / z) <= 1e+76)
		tmp = Float64(Float64(y / Float64(z * x_m)) * fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * 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[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+76], N[(N[(y / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e76

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.2

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 82.2

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

   7. Applied rewrites 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if 1e76 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 77.8%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 82.1

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 82.1

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

   7. Applied rewrites 82.1%

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

Alternative 4: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x\_m \cdot \frac{y}{x\_m}}{z} \leq 1000:\\ \;\;\;\;\frac{t\_0}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{z}}{x\_m}\\ \end{array} \end{array} \end{array} \]

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 0.5 (* x_m x_m) 1.0) y)))
   (*
    x_s
    (if (<= (/ (* (cosh x_m) (/ y x_m)) z) 1000.0)
      (/ t_0 (* z x_m))
      (/ (/ t_0 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(0.5, (x_m * x_m), 1.0) * y;
	double tmp;
	if (((cosh(x_m) * (y / x_m)) / z) <= 1000.0) {
		tmp = t_0 / (z * x_m);
	} else {
		tmp = (t_0 / 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 = Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y / x_m)) / z) <= 1000.0)
		tmp = Float64(t_0 / Float64(z * x_m));
	else
		tmp = Float64(Float64(t_0 / 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[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1000.0], N[(t$95$0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1e3

   1. Initial program 97.7%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 67.8

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 67.8

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

   7. Applied rewrites 67.8%

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

   if 1e3 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.7%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 82.8

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 82.8

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

   7. Applied rewrites 82.8%

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

Alternative 5: 73.9% accurate, 0.7× speedup

Math

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

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 (<= (/ (* (cosh x_m) (/ y x_m)) z) 1e+32)
    (/ (* (fma 0.5 (* x_m x_m) 1.0) y) (* z x_m))
    (* (/ (fma (* x_m x_m) 0.5 1.0) x_m) (/ y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((cosh(x_m) * (y / x_m)) / z) <= 1e+32) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y) / (z * x_m);
	} else {
		tmp = (fma((x_m * x_m), 0.5, 1.0) / x_m) * (y / z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x_m) * Float64(y / x_m)) / z) <= 1e+32)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(fma(Float64(x_m * x_m), 0.5, 1.0) / x_m) * Float64(y / 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[N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+32], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000005e32

   1. Initial program 97.8%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 68.7

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 68.7

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

   7. Applied rewrites 68.7%

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

   if 1.00000000000000005e32 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 78.0%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 72.7%

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

Alternative 6: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y}{x\_m} \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{z \cdot x\_m}\\ \end{array} \end{array} \]

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 (<= (* (cosh x_m) (/ y x_m)) 5e+81)
    (/ (/ y x_m) z)
    (/ (* (fma 0.5 (* x_m x_m) 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 ((cosh(x_m) * (y / x_m)) <= 5e+81) {
		tmp = (y / x_m) / z;
	} else {
		tmp = (fma(0.5, (x_m * x_m), 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(cosh(x_m) * Float64(y / x_m)) <= 5e+81)
		tmp = Float64(Float64(y / x_m) / z);
	else
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / Float64(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[Cosh[x$95$m], $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision], 5e+81], N[(N[(y / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 4.9999999999999998e81

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 4.9999999999999998e81 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 77.0%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 65.8

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 65.8

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

   7. Applied rewrites 65.8%

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

Alternative 7: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 3.5 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \frac{\cosh x\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \end{array} \end{array} \]

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 2e-18)
    (/ (/ y z) x_m)
    (if (<= x_m 3.5e+45)
      (* y (/ (cosh x_m) (* z x_m)))
      (/
       (*
        (/
         (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)
        y)
       z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2e-18) {
		tmp = (y / z) / x_m;
	} else if (x_m <= 3.5e+45) {
		tmp = y * (cosh(x_m) / (z * x_m));
	} else {
		tmp = ((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) * y) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2e-18)
		tmp = Float64(Float64(y / z) / x_m);
	elseif (x_m <= 3.5e+45)
		tmp = Float64(y * Float64(cosh(x_m) / Float64(z * x_m)));
	else
		tmp = Float64(Float64(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) * y) / 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, 2e-18], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 3.5e+45], N[(y * N[(N[Cosh[x$95$m], $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(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] * y), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.0000000000000001e-18

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 2.0000000000000001e-18 < x < 3.50000000000000023e45

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 3.50000000000000023e45 < x

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 100.0%

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

Alternative 8: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, 0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right) \cdot x\_m, x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x\_m, x\_m, 0.5\right), y\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

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 2e+101)
    (/
     (/
      (*
       y
       (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)
     z)
    (/
     (/ (fma (* (* x_m x_m) y) (fma (* 0.041666666666666664 x_m) x_m 0.5) 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 <= 2e+101) {
		tmp = ((y * 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) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y), fma((0.041666666666666664 * x_m), x_m, 0.5), 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 (y <= 2e+101)
		tmp = Float64(Float64(Float64(y * 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) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y), fma(Float64(0.041666666666666664 * x_m), x_m, 0.5), 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[y, 2e+101], N[(N[(N[(y * 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]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y), $MachinePrecision] * N[(N[(0.041666666666666664 * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2e101

   1. Initial program 87.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-/.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

   13. Applied rewrites 89.7%

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

   if 2e101 < y

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 92.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x, x, 0.5\right) \cdot x, x, 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x\_m, x\_m, 0.5\right), y\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

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 1.8e+101)
    (/
     (*
      (/
       (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)
      y)
     z)
    (/
     (/ (fma (* (* x_m x_m) y) (fma (* 0.041666666666666664 x_m) x_m 0.5) y) z)
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.8e+101) {
		tmp = ((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) * y) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y), fma((0.041666666666666664 * x_m), x_m, 0.5), 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 (y <= 1.8e+101)
		tmp = Float64(Float64(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) * y) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y), fma(Float64(0.041666666666666664 * x_m), x_m, 0.5), 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[y, 1.8e+101], N[(N[(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] * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y), $MachinePrecision] * N[(N[(0.041666666666666664 * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000015e101

   1. Initial program 87.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-/.f64 N/A

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 89.6%

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

   if 1.80000000000000015e101 < y

   1. Initial program 91.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 92.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}{x} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 7 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.041666666666666664, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{z}}{x\_m}\\ \end{array} \end{array} \]

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 (<= x_m 1.86e-56)
    (/ (/ y z) x_m)
    (if (<= x_m 7e+130)
      (/
       (* (fma (fma (* x_m x_m) 0.041666666666666664 0.5) (* x_m x_m) 1.0) y)
       (* z x_m))
      (/ (/ (* (fma 0.5 (* x_m x_m) 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 (x_m <= 1.86e-56) {
		tmp = (y / z) / x_m;
	} else if (x_m <= 7e+130) {
		tmp = (fma(fma((x_m * x_m), 0.041666666666666664, 0.5), (x_m * x_m), 1.0) * y) / (z * x_m);
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 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 (x_m <= 1.86e-56)
		tmp = Float64(Float64(y / z) / x_m);
	elseif (x_m <= 7e+130)
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.041666666666666664, 0.5), Float64(x_m * x_m), 1.0) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / z) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.86e-56], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 7e+130], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664 + 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[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.85999999999999997e-56

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 1.85999999999999997e-56 < x < 7.0000000000000002e130

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 59.6

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 59.6

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

   7. Applied rewrites 59.6%

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

   if 7.0000000000000002e130 < x

   1. Initial program 81.3%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 96.9

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 96.9

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

   7. Applied rewrites 96.9%

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

Alternative 11: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1500000000000:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x\_m \cdot x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x\_m, x\_m, 0.5\right), y\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

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 1500000000000.0)
    (/
     (/
      (* y (fma (fma 0.041666666666666664 (* x_m x_m) 0.5) (* x_m x_m) 1.0))
      x_m)
     z)
    (/
     (/ (fma (* (* x_m x_m) y) (fma (* 0.041666666666666664 x_m) x_m 0.5) 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 <= 1500000000000.0) {
		tmp = ((y * fma(fma(0.041666666666666664, (x_m * x_m), 0.5), (x_m * x_m), 1.0)) / x_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y), fma((0.041666666666666664 * x_m), x_m, 0.5), 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 (y <= 1500000000000.0)
		tmp = Float64(Float64(Float64(y * fma(fma(0.041666666666666664, Float64(x_m * x_m), 0.5), Float64(x_m * x_m), 1.0)) / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y), fma(Float64(0.041666666666666664 * x_m), x_m, 0.5), 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[y, 1500000000000.0], N[(N[(N[(y * N[(N[(0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y), $MachinePrecision] * N[(N[(0.041666666666666664 * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e12

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-/.f64 N/A

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

   5. Applied rewrites 86.0%

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

   if 1.5e12 < y

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 92.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1500000000000:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1200000000000:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664, x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x\_m, x\_m, 0.5\right), y\right)}{z}}{x\_m}\\ \end{array} \end{array} \]

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 1200000000000.0)
    (/
     (/ (* y (fma (* (* x_m x_m) 0.041666666666666664) (* x_m x_m) 1.0)) x_m)
     z)
    (/
     (/ (fma (* (* x_m x_m) y) (fma (* 0.041666666666666664 x_m) x_m 0.5) 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 <= 1200000000000.0) {
		tmp = ((y * fma(((x_m * x_m) * 0.041666666666666664), (x_m * x_m), 1.0)) / x_m) / z;
	} else {
		tmp = (fma(((x_m * x_m) * y), fma((0.041666666666666664 * x_m), x_m, 0.5), 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 (y <= 1200000000000.0)
		tmp = Float64(Float64(Float64(y * fma(Float64(Float64(x_m * x_m) * 0.041666666666666664), Float64(x_m * x_m), 1.0)) / x_m) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_m * x_m) * y), fma(Float64(0.041666666666666664 * x_m), x_m, 0.5), 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[y, 1200000000000.0], N[(N[(N[(y * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y), $MachinePrecision] * N[(N[(0.041666666666666664 * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e12

   1. Initial program 86.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. lower-/.f64 N/A

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.6

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

   8. Applied rewrites 85.6%

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

   if 1.2e12 < y

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 92.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1200000000000:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right), y\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.041666666666666664, x\_m \cdot x\_m, 1\right)}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{z}}{x\_m}\\ \end{array} \end{array} \]

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 4.4e+51)
    (/
     (/ (* y (fma (* (* x_m x_m) 0.041666666666666664) (* x_m x_m) 1.0)) x_m)
     z)
    (/ (/ (* (fma 0.5 (* x_m x_m) 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 <= 4.4e+51) {
		tmp = ((y * fma(((x_m * x_m) * 0.041666666666666664), (x_m * x_m), 1.0)) / x_m) / z;
	} else {
		tmp = ((fma(0.5, (x_m * x_m), 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 (y <= 4.4e+51)
		tmp = Float64(Float64(Float64(y * fma(Float64(Float64(x_m * x_m) * 0.041666666666666664), Float64(x_m * x_m), 1.0)) / x_m) / z);
	else
		tmp = Float64(Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / 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[y, 4.4e+51], N[(N[(N[(y * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999984e51

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.1

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

   8. Applied rewrites 86.1%

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

   if 4.39999999999999984e51 < y

   1. Initial program 92.3%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.0

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 91.0

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

   7. Applied rewrites 91.0%

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

Alternative 14: 76.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x\_m \cdot x\_m, 1\right) \cdot y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x\_m \cdot x\_m\right) \cdot 0.5}{x\_m} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

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 1.86e-56)
    (/ (/ y z) x_m)
    (if (<= x_m 4e+143)
      (/ (* (fma 0.5 (* x_m x_m) 1.0) y) (* z x_m))
      (* (/ (* (* x_m x_m) 0.5) x_m) (/ y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.86e-56) {
		tmp = (y / z) / x_m;
	} else if (x_m <= 4e+143) {
		tmp = (fma(0.5, (x_m * x_m), 1.0) * y) / (z * x_m);
	} else {
		tmp = (((x_m * x_m) * 0.5) / x_m) * (y / z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.86e-56)
		tmp = Float64(Float64(y / z) / x_m);
	elseif (x_m <= 4e+143)
		tmp = Float64(Float64(fma(0.5, Float64(x_m * x_m), 1.0) * y) / Float64(z * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * 0.5) / x_m) * Float64(y / 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, 1.86e-56], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], If[LessEqual[x$95$m, 4e+143], N[(N[(N[(0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.85999999999999997e-56

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 1.85999999999999997e-56 < x < 4.0000000000000001e143

   1. Initial program 94.6%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 39.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 43.0

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 43.0

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

   7. Applied rewrites 43.0%

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

   if 4.0000000000000001e143 < x

   1. Initial program 78.6%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.9

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

   8. Applied rewrites 92.9%

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

Alternative 15: 62.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;\frac{\frac{y}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{0.5}{z}\right) \cdot x\_m\\ \end{array} \end{array} \]

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 (<= x_m 0.9) (/ (/ y z) x_m) (* (* y (/ 0.5 z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (y / z) / x_m;
	} else {
		tmp = (y * (0.5 / z)) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_s, x_m, y, z)
use fmin_fmax_functions
    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 <= 0.9d0) then
        tmp = (y / z) / x_m
    else
        tmp = (y * (0.5d0 / z)) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = (y / z) / x_m;
	} else {
		tmp = (y * (0.5 / z)) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 0.9:
		tmp = (y / z) / x_m
	else:
		tmp = (y * (0.5 / z)) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(Float64(y / z) / x_m);
	else
		tmp = Float64(Float64(y * Float64(0.5 / z)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = (y / z) / x_m;
	else
		tmp = (y * (0.5 / z)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 0.9], N[(N[(y / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(y * N[(0.5 / z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 0.900000000000000022 < x

   1. Initial program 88.6%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 36.9

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

   8. Applied rewrites 36.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 36.9

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

   11. Applied rewrites 36.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

          \[\leadsto \frac{y \cdot \frac{1}{2}}{z} \cdot x \]

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 36.9

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

   13. Applied rewrites 36.9%

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

Alternative 16: 62.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.9:\\ \;\;\;\;\frac{y}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{0.5}{z}\right) \cdot x\_m\\ \end{array} \end{array} \]

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 (<= x_m 0.9) (/ y (* z x_m)) (* (* y (/ 0.5 z)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = y / (z * x_m);
	} else {
		tmp = (y * (0.5 / z)) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_s, x_m, y, z)
use fmin_fmax_functions
    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 <= 0.9d0) then
        tmp = y / (z * x_m)
    else
        tmp = (y * (0.5d0 / z)) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 0.9) {
		tmp = y / (z * x_m);
	} else {
		tmp = (y * (0.5 / z)) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 0.9:
		tmp = y / (z * x_m)
	else:
		tmp = (y * (0.5 / z)) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 0.9)
		tmp = Float64(y / Float64(z * x_m));
	else
		tmp = Float64(Float64(y * Float64(0.5 / z)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 0.9)
		tmp = y / (z * x_m);
	else
		tmp = (y * (0.5 / z)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 0.9], N[(y / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(0.5 / z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 61.8

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

   7. Applied rewrites 61.8%

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

   if 0.900000000000000022 < x

   1. Initial program 88.6%

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

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

      2. div-add-rev N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt1-in N/A

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

      12. +-commutative N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-/r* N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 36.9

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

   8. Applied rewrites 36.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 36.9

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

   11. Applied rewrites 36.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

          \[\leadsto \frac{y \cdot \frac{1}{2}}{z} \cdot x \]

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 36.9

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

   13. Applied rewrites 36.9%

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

Alternative 17: 49.7% 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 =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_s, x_m, y, z)
use fmin_fmax_functions
    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 88.4%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 54.1

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

5. Applied rewrites 54.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 47.3

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

7. Applied rewrites 47.3%

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

Developer Target 1: 97.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025026 
(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))