Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.1% → 97.4%

Time: 4.6s

Alternatives: 16

Speedup: 0.8×

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

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

Fortran

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: 84.1% 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: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.5 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 5.5e+191)
    (/ (/ (* (cosh x) y_m) x) z)
    (/ (/ (fma (* (* x x) y_m) 0.5 y_m) z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5.5e+191) {
		tmp = ((cosh(x) * y_m) / x) / z;
	} else {
		tmp = (fma(((x * x) * y_m), 0.5, y_m) / z) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 5.5e+191)
		tmp = Float64(Float64(Float64(cosh(x) * y_m) / x) / z);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 0.5, y_m) / z) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 5.5e+191], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000002e191

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cosh.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. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cosh.f64 97.6

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

   3. Applied rewrites 97.6%

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

   if 5.5000000000000002e191 < y

   1. Initial program 87.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 2: 94.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (Math.cosh(x) * (y_m / x)) / z;
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (y_m * (((x * x) * 0.5) / z)) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (math.cosh(x) * (y_m / x)) / z
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (y_m * (((x * x) * 0.5) / z)) / x
	return y_s * tmp

Julia

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

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (cosh(x) * (y_m / x)) / z;
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (y_m * (((x * x) * 0.5) / z)) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $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 95.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 87.0

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

   6. Applied rewrites 87.0%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 87.0

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

   9. Applied rewrites 87.0%

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

Alternative 3: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \frac{1}{z}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;\frac{\cosh x \cdot y\_m}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 7.5e-192)
    (* (/ y_m x) (/ 1.0 z))
    (if (<= x 2.3e+151)
      (/ (* (cosh x) y_m) (* z x))
      (/ (* y_m (/ (* (* x x) 0.5) z)) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.5e-192) {
		tmp = (y_m / x) * (1.0 / z);
	} else if (x <= 2.3e+151) {
		tmp = (cosh(x) * y_m) / (z * x);
	} else {
		tmp = (y_m * (((x * x) * 0.5) / z)) / x;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 7.5d-192) then
        tmp = (y_m / x) * (1.0d0 / z)
    else if (x <= 2.3d+151) then
        tmp = (cosh(x) * y_m) / (z * x)
    else
        tmp = (y_m * (((x * x) * 0.5d0) / z)) / x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.5e-192) {
		tmp = (y_m / x) * (1.0 / z);
	} else if (x <= 2.3e+151) {
		tmp = (Math.cosh(x) * y_m) / (z * x);
	} else {
		tmp = (y_m * (((x * x) * 0.5) / z)) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 7.5e-192:
		tmp = (y_m / x) * (1.0 / z)
	elif x <= 2.3e+151:
		tmp = (math.cosh(x) * y_m) / (z * x)
	else:
		tmp = (y_m * (((x * x) * 0.5) / z)) / x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 7.5e-192)
		tmp = Float64(Float64(y_m / x) * Float64(1.0 / z));
	elseif (x <= 2.3e+151)
		tmp = Float64(Float64(cosh(x) * y_m) / Float64(z * x));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(Float64(x * x) * 0.5) / z)) / x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 7.5e-192)
		tmp = (y_m / x) * (1.0 / z);
	elseif (x <= 2.3e+151)
		tmp = (cosh(x) * y_m) / (z * x);
	else
		tmp = (y_m * (((x * x) * 0.5) / z)) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 7.5e-192], N[(N[(y$95$m / x), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e+151], N[(N[(N[Cosh[x], $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 7.5000000000000001e-192

   1. Initial program 84.8%

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

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cosh.f64 84.8

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 54.9%

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

   if 7.5000000000000001e-192 < x < 2.3000000000000001e151

   1. Initial program 91.9%

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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.2

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

   3. Applied rewrites 92.2%

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

   if 2.3000000000000001e151 < x

   1. Initial program 64.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 4: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{t\_0}{z}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.5 1.0)))
   (*
    y_s
    (if (<= (* (cosh x) (/ y_m x)) 5e+135)
      (/ (* t_0 (/ y_m x)) z)
      (/ (* y_m (/ t_0 z)) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), 0.5, 1.0);
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 5e+135) {
		tmp = (t_0 * (y_m / x)) / z;
	} else {
		tmp = (y_m * (t_0 / z)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 5e+135)
		tmp = Float64(Float64(t_0 * Float64(y_m / x)) / z);
	else
		tmp = Float64(Float64(y_m * Float64(t_0 / z)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 5e+135], N[(N[(t$95$0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000029e135

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   if 5.00000000000000029e135 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 81.6

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

   6. Applied rewrites 81.6%

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

Alternative 5: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y\_m, 0.5, y\_m\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 5e+135)
    (/ (/ (fma (* (* x x) y_m) 0.5 y_m) x) z)
    (/ (* y_m (/ (fma (* x x) 0.5 1.0) z)) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 5e+135) {
		tmp = (fma(((x * x) * y_m), 0.5, y_m) / x) / z;
	} else {
		tmp = (y_m * (fma((x * x), 0.5, 1.0) / z)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 5e+135)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y_m), 0.5, y_m) / x) / z);
	else
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), 0.5, 1.0) / z)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 5e+135], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.5 + y$95$m), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000029e135

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

   if 5.00000000000000029e135 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 81.6

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

   6. Applied rewrites 81.6%

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

Alternative 6: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y\_m}{x}}{z} \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{t\_0}{z \cdot x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{t\_0}{z}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.5 1.0)))
   (*
    y_s
    (if (<= (/ (* (cosh x) (/ y_m x)) z) 5e-49)
      (* (/ t_0 (* z x)) y_m)
      (/ (* y_m (/ t_0 z)) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = fma((x * x), 0.5, 1.0);
	double tmp;
	if (((cosh(x) * (y_m / x)) / z) <= 5e-49) {
		tmp = (t_0 / (z * x)) * y_m;
	} else {
		tmp = (y_m * (t_0 / z)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 5e-49)
		tmp = Float64(Float64(t_0 / Float64(z * x)) * y_m);
	else
		tmp = Float64(Float64(y_m * Float64(t_0 / z)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e-49], N[(N[(t$95$0 / N[(z * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(y$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.9999999999999999e-49

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.7

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

   4. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

   if 4.9999999999999999e-49 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 83.0

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

   6. Applied rewrites 83.0%

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

Alternative 7: 71.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 2 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 2e+197)
    (/ (/ y_m x) z)
    (* y_m (/ (/ (fma (* x x) 0.5 1.0) z) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 2e+197) {
		tmp = (y_m / x) / z;
	} else {
		tmp = y_m * ((fma((x * x), 0.5, 1.0) / z) / x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 2e+197)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(y_m * Float64(Float64(fma(Float64(x * x), 0.5, 1.0) / z) / x));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 2e+197], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999999e197

   1. Initial program 96.3%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 65.5

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

   4. Applied rewrites 65.5%

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

   if 1.9999999999999999e197 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 79.1

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

   8. Applied rewrites 79.1%

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

Alternative 8: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1000000.0)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (/ (* y_m (/ (* (* x x) 0.5) z)) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1000000.0) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else {
		tmp = (y_m * (((x * x) * 0.5) / z)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1000000.0)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(Float64(x * x) * 0.5) / z)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1000000.0], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e6

   1. Initial program 86.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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

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

   6. Applied rewrites 79.8%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 72.1

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

   9. Applied rewrites 72.1%

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

   if 1e6 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 72.2

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

   9. Applied rewrites 72.2%

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

Alternative 9: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 250000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+198}:\\ \;\;\;\;y\_m \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 250000.0)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (if (<= x 6.5e+198)
      (* y_m (/ (* (* x x) 0.5) (* z x)))
      (/ (* (* 0.5 x) y_m) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 250000.0) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else if (x <= 6.5e+198) {
		tmp = y_m * (((x * x) * 0.5) / (z * x));
	} else {
		tmp = ((0.5 * x) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 250000.0)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	elseif (x <= 6.5e+198)
		tmp = Float64(y_m * Float64(Float64(Float64(x * x) * 0.5) / Float64(z * x)));
	else
		tmp = Float64(Float64(Float64(0.5 * x) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 250000.0], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e+198], N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.5e5

   1. Initial program 86.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 72.2

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

   9. Applied rewrites 72.2%

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

   if 2.5e5 < x < 6.5000000000000003e198

   1. Initial program 85.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 35.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.9

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

   9. Applied rewrites 35.9%

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

   if 6.5000000000000003e198 < x

   1. Initial program 60.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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

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

   6. Applied rewrites 89.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 44.0

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

   9. Applied rewrites 44.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 56.2

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

   11. Applied rewrites 56.2%

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

Alternative 10: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 420000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \frac{1}{x}\right) \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot y\_m}{z}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 420000.0)
    (* (fma 0.5 x (/ 1.0 x)) (/ y_m z))
    (/ (* (* 0.5 x) y_m) z))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 420000.0) {
		tmp = fma(0.5, x, (1.0 / x)) * (y_m / z);
	} else {
		tmp = ((0.5 * x) * y_m) / z;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 420000.0)
		tmp = Float64(fma(0.5, x, Float64(1.0 / x)) * Float64(y_m / z));
	else
		tmp = Float64(Float64(Float64(0.5 * x) * y_m) / z);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 420000.0], N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e5

   1. Initial program 86.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. pow-flip N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 72.1

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

   9. Applied rewrites 72.1%

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

   if 4.2e5 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 32.4

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

   9. Applied rewrites 32.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 41.2

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

   11. Applied rewrites 41.2%

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

Alternative 11: 57.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (y_m / x) / z
    else
        tmp = ((0.5d0 * x) * y_m) / z
    end if
    code = y_s * tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 63.0

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

   4. Applied rewrites 63.0%

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

   if 1.3999999999999999 < x

   1. Initial program 76.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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

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

   6. Applied rewrites 56.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 32.3

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

   9. Applied rewrites 32.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 40.9

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

   11. Applied rewrites 40.9%

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

Alternative 12: 56.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (y_m / x) / z
    else
        tmp = y_m * ((x / z) * 0.5d0)
    end if
    code = y_s * tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 63.0

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

   4. Applied rewrites 63.0%

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

   if 1.3999999999999999 < x

   1. Initial program 76.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.6

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

   4. Applied rewrites 70.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-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. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 44.5%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 39.4

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

   9. Applied rewrites 39.4%

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

Alternative 13: 55.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (y_m / x) / z
    else
        tmp = (x * (y_m / z)) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 63.0

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

   4. Applied rewrites 63.0%

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

   if 1.3999999999999999 < x

   1. Initial program 76.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.6

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 32.3

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

   7. Applied rewrites 32.3%

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

Alternative 14: 52.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y\_m}{x} \leq 10^{+107}:\\ \;\;\;\;\frac{\frac{y\_m}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (cosh x) (/ y_m x)) 1e+107) (/ (/ y_m x) z) (/ (/ y_m z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((cosh(x) * (y_m / x)) <= 1e+107) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((cosh(x) * (y_m / x)) <= 1d+107) then
        tmp = (y_m / x) / z
    else
        tmp = (y_m / z) / x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((Math.cosh(x) * (y_m / x)) <= 1e+107) {
		tmp = (y_m / x) / z;
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if (math.cosh(x) * (y_m / x)) <= 1e+107:
		tmp = (y_m / x) / z
	else:
		tmp = (y_m / z) / x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y_m / x)) <= 1e+107)
		tmp = Float64(Float64(y_m / x) / z);
	else
		tmp = Float64(Float64(y_m / z) / x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((cosh(x) * (y_m / x)) <= 1e+107)
		tmp = (y_m / x) / z;
	else
		tmp = (y_m / z) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], 1e+107], N[(N[(y$95$m / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999997e106

   1. Initial program 96.1%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

   if 9.9999999999999997e106 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 70.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 39.8%

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

Alternative 15: 52.5% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (cosh x) (/ y_m x)) z) 1e-50)
    (/ y_m (* z x))
    (/ (/ y_m z) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((cosh(x) * (y_m / x)) / z) <= 1e-50) {
		tmp = y_m / (z * x);
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x) * (y_m / x)) / z) <= 1d-50) then
        tmp = y_m / (z * x)
    else
        tmp = (y_m / z) / x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((Math.cosh(x) * (y_m / x)) / z) <= 1e-50) {
		tmp = y_m / (z * x);
	} else {
		tmp = (y_m / z) / x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if ((math.cosh(x) * (y_m / x)) / z) <= 1e-50:
		tmp = y_m / (z * x)
	else:
		tmp = (y_m / z) / x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y_m / x)) / z) <= 1e-50)
		tmp = Float64(y_m / Float64(z * x));
	else
		tmp = Float64(Float64(y_m / z) / x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (((cosh(x) * (y_m / x)) / z) <= 1e-50)
		tmp = y_m / (z * x);
	else
		tmp = (y_m / z) / x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.00000000000000001e-50

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   if 1.00000000000000001e-50 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 72.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 45.6%

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

Alternative 16: 49.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

y\_m =     private
y\_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(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m / (z * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 49.2

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

4. Applied rewrites 49.2%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))