Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 94.2%

Time: 3.8s

Alternatives: 15

Speedup: 0.9×

• 63.7% 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: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 94.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 INFINITY) t_0 (/ (* (/ (* (* x x) 0.5) z) y) x))))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = ((((x * x) * 0.5) / z) * y) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 0.5) / z) * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / x), $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 80.2

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

   4. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 86.6

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

   9. Applied rewrites 86.6%

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

Alternative 2: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z} \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.4e+129)
   (/ (* (cosh x) y) (* z x))
   (/ (* (/ (fma (* x x) 0.5 1.0) z) y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.4e+129) {
		tmp = (cosh(x) * y) / (z * x);
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) / z) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.4e+129)
		tmp = Float64(Float64(cosh(x) * y) / Float64(z * x));
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) / z) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.4e+129], N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.39999999999999986e129

   1. Initial program 87.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 87.5

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

   3. Applied rewrites 87.5%

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

   if 8.39999999999999986e129 < x

   1. Initial program 66.8%

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

   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 94.1

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

   4. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 94.5

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

   6. Applied rewrites 94.5%

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

Alternative 3: 81.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{y}{x}}{z}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot 0.5}{z} \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 2e+58)
     (/ (* (fma (* x x) 0.5 1.0) (/ y x)) z)
     (if (<= t_0 INFINITY)
       (/ (/ (fma (* (* x x) y) 0.5 y) z) x)
       (/ (* (/ (* (* x x) 0.5) z) y) x)))))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= 2e+58) {
		tmp = (fma((x * x), 0.5, 1.0) * (y / x)) / z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fma(((x * x) * y), 0.5, y) / z) / x;
	} else {
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= 2e+58)
		tmp = Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * Float64(y / x)) / z);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y), 0.5, y) / z) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 0.5) / z) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+58], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999989e58

   1. Initial program 96.3%

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

   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 80.8

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

   4. Applied rewrites 80.8%

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

   if 1.99999999999999989e58 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 93.3%

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

   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 81.8

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

   4. Applied rewrites 81.8%

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

   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 80.2

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

   4. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 86.6

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

   9. Applied rewrites 86.6%

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

Alternative 4: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot 0.5}{z} \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (<= t_0 2e+58)
     (/ (/ y x) z)
     (if (<= t_0 INFINITY)
       (/ (/ (fma (* (* x x) y) 0.5 y) z) x)
       (/ (* (/ (* (* x x) 0.5) z) y) x)))))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if (t_0 <= 2e+58) {
		tmp = (y / x) / z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fma(((x * x) * y), 0.5, y) / z) / x;
	} else {
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if (t_0 <= 2e+58)
		tmp = Float64(Float64(y / x) / z);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y), 0.5, y) / z) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 0.5) / z) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+58], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999989e58

   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 62.5

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

   4. Applied rewrites 62.5%

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

   if 1.99999999999999989e58 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 93.3%

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

   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 81.8

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

   4. Applied rewrites 81.8%

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

   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 80.2

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

   4. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 86.6

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

   9. Applied rewrites 86.6%

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

Alternative 5: 71.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)}{z} \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (cosh x) (/ y x)) z) 2e+58)
   (/ (/ y x) z)
   (/ (* (/ (fma (* x x) 0.5 1.0) z) y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((cosh(x) * (y / x)) / z) <= 2e+58) {
		tmp = (y / x) / z;
	} else {
		tmp = ((fma((x * x), 0.5, 1.0) / z) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y / x)) / z) <= 2e+58)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.5, 1.0) / z) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+58], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999989e58

   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 62.5

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

   4. Applied rewrites 62.5%

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

   if 1.99999999999999989e58 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.3%

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

   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 81.4

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

   4. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 6: 67.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4) (/ (/ y z) x) (/ (* (/ (* (* x x) 0.5) z) y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = ((((x * x) * 0.5) / z) * y) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 0.5) / z) * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (y / z) / x;
	else
		tmp = ((((x * x) * 0.5) / z) * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.0%

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

   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 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. Taylor expanded in x around 0

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

   7. Applied rewrites 67.1%

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

   if 1.3999999999999999 < x

   1. Initial program 78.3%

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

   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 70.5

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

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, y\right)}{z}}{x}} \]
   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. *-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. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. +-commutative N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 70.2

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

   6. Applied rewrites 70.2%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 70.2

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

   9. Applied rewrites 70.2%

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

Alternative 7: 62.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(\left(0.5 \cdot x\right) \cdot y\right) \cdot x}{z \cdot x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right)}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4)
   (/ (/ y z) x)
   (if (<= x 7.5e+113)
     (/ (* (* (* 0.5 x) y) x) (* z x))
     (if (<= x 8.2e+265)
       (* (/ (* (* x x) 0.5) (* z x)) y)
       (* (/ (fma 0.5 x (/ 1.0 x)) z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if (x <= 7.5e+113) {
		tmp = (((0.5 * x) * y) * x) / (z * x);
	} else if (x <= 8.2e+265) {
		tmp = (((x * x) * 0.5) / (z * x)) * y;
	} else {
		tmp = (fma(0.5, x, (1.0 / x)) / z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 7.5e+113)
		tmp = Float64(Float64(Float64(Float64(0.5 * x) * y) * x) / Float64(z * x));
	elseif (x <= 8.2e+265)
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) / Float64(z * x)) * y);
	else
		tmp = Float64(Float64(fma(0.5, x, Float64(1.0 / x)) / z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.5e+113], N[(N[(N[(N[(0.5 * x), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e+265], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.0%

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

   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 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. Taylor expanded in x around 0

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

   7. Applied rewrites 67.1%

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

   if 1.3999999999999999 < x < 7.5000000000000001e113

   1. Initial program 95.6%

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

   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 34.3

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 34.3

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

   7. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(z \cdot x\right)\right)} \]

   9. Applied rewrites 29.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. pow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 29.1

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

   11. Applied rewrites 29.1%

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

   if 7.5000000000000001e113 < x < 8.2000000000000007e265

   1. Initial program 72.7%

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

   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 89.3

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

   4. Applied rewrites 89.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      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

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 59.4%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if 8.2000000000000007e265 < x

   1. Initial program 51.0%

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

   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 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

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

      2. lift-/.f64 N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      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

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. pow-flip N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. inv-pow N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 58.1

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

   11. Applied rewrites 58.1%

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

Alternative 8: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right)}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4)
   (/ (/ y z) x)
   (if (<= x 8.2e+265)
     (/ (* (* (* x x) 0.5) y) (* z x))
     (* (/ (fma 0.5 x (/ 1.0 x)) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if (x <= 8.2e+265) {
		tmp = (((x * x) * 0.5) * y) / (z * x);
	} else {
		tmp = (fma(0.5, x, (1.0 / x)) / z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 8.2e+265)
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) * y) / Float64(z * x));
	else
		tmp = Float64(Float64(fma(0.5, x, Float64(1.0 / x)) / z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.2e+265], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.0%

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

   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 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. Taylor expanded in x around 0

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

   7. Applied rewrites 67.1%

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

   if 1.3999999999999999 < x < 8.2000000000000007e265

   1. Initial program 82.3%

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

   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 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot y}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(z \cdot x\right)\right)} \]

   9. Applied rewrites 47.0%

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

   if 8.2000000000000007e265 < x

   1. Initial program 51.0%

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

   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 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

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

      2. lift-/.f64 N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      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

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. pow-flip N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. inv-pow N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 58.1

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

   11. Applied rewrites 58.1%

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

Alternative 9: 61.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{z \cdot x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right)}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4)
   (/ (/ y z) x)
   (if (<= x 8.2e+265)
     (* (/ (* (* x x) 0.5) (* z x)) y)
     (* (/ (fma 0.5 x (/ 1.0 x)) z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if (x <= 8.2e+265) {
		tmp = (((x * x) * 0.5) / (z * x)) * y;
	} else {
		tmp = (fma(0.5, x, (1.0 / x)) / z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 8.2e+265)
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) / Float64(z * x)) * y);
	else
		tmp = Float64(Float64(fma(0.5, x, Float64(1.0 / x)) / z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.2e+265], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.0%

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

   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 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. Taylor expanded in x around 0

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

   7. Applied rewrites 67.1%

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

   if 1.3999999999999999 < x < 8.2000000000000007e265

   1. Initial program 82.3%

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

   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 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      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

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 45.2%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 45.2

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

   9. Applied rewrites 45.2%

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

   if 8.2000000000000007e265 < x

   1. Initial program 51.0%

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

   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 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

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

      2. lift-/.f64 N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      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

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. pow-flip N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. inv-pow N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 58.1

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

   11. Applied rewrites 58.1%

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

Alternative 10: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, \frac{1}{x}\right)}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (cosh x) (/ y x)) z) 5e+306)
   (/ (/ y x) z)
   (* (/ (fma 0.5 x (/ 1.0 x)) z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((cosh(x) * (y / x)) / z) <= 5e+306) {
		tmp = (y / x) / z;
	} else {
		tmp = (fma(0.5, x, (1.0 / x)) / z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y / x)) / z) <= 5e+306)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(fma(0.5, x, Float64(1.0 / x)) / z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 5e+306], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(0.5 * x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.99999999999999993e306

   1. Initial program 96.7%

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

   2. Taylor expanded in x around 0

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

      1. lift-/.f64 66.3

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

   4. Applied rewrites 66.3%

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

   if 4.99999999999999993e306 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 66.6%

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

   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 78.7

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

   4. Applied rewrites 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.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. lift-*.f64 N/A

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

      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

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      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. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. pow-flip N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. inv-pow N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 54.1

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

   11. Applied rewrites 54.1%

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

Alternative 11: 60.3% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4) (/ (/ y z) x) (/ (* (* 0.5 x) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = ((0.5 * x) * y) / z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = ((0.5 * x) * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = ((0.5 * x) * y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(Float64(0.5 * x) * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (y / z) / x;
	else
		tmp = ((0.5 * x) * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.0%

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

   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 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. Taylor expanded in x around 0

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

   7. Applied rewrites 67.1%

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

   if 1.3999999999999999 < x

   1. Initial program 78.3%

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

   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 70.5

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

   4. Applied rewrites 70.5%

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

   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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.3

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

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 40.4

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

   9. Applied rewrites 40.4%

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

Alternative 12: 58.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4) (/ (/ y z) x) (* (* (/ y z) x) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = ((y / z) * x) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = ((y / z) * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = ((y / z) * x) * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(Float64(y / z) * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (y / z) / x;
	else
		tmp = ((y / z) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 87.0%

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

   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 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. Taylor expanded in x around 0

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

   7. Applied rewrites 67.1%

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

   if 1.3999999999999999 < x

   1. Initial program 78.3%

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

   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 70.5

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

   4. Applied rewrites 70.5%

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

   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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 32.3

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

   7. Applied rewrites 32.3%

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

Alternative 13: 53.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (cosh x) (/ y x)) z) 2e+58) (/ (/ y x) z) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((cosh(x) * (y / x)) / z) <= 2e+58) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x) * (y / x)) / z) <= 2d+58) then
        tmp = (y / x) / z
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((Math.cosh(x) * (y / x)) / z) <= 2e+58) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((math.cosh(x) * (y / x)) / z) <= 2e+58:
		tmp = (y / x) / z
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y / x)) / z) <= 2e+58)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((cosh(x) * (y / x)) / z) <= 2e+58)
		tmp = (y / x) / z;
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+58], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999989e58

   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 62.5

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

   4. Applied rewrites 62.5%

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

   if 1.99999999999999989e58 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.3%

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

   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 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 42.7%

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

Alternative 14: 53.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* (cosh x) (/ y x)) z) 2e+37) (/ y (* z x)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((cosh(x) * (y / x)) / z) <= 2e+37) {
		tmp = y / (z * x);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((cosh(x) * (y / x)) / z) <= 2d+37) then
        tmp = y / (z * x)
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((Math.cosh(x) * (y / x)) / z) <= 2e+37) {
		tmp = y / (z * x);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((math.cosh(x) * (y / x)) / z) <= 2e+37:
		tmp = y / (z * x)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y / x)) / z) <= 2e+37)
		tmp = Float64(y / Float64(z * x));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((cosh(x) * (y / x)) / z) <= 2e+37)
		tmp = y / (z * x);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e+37], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999991e37

   1. Initial program 96.3%

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

   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 62.2

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

   4. Applied rewrites 62.2%

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

   if 1.99999999999999991e37 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 71.7%

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

   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 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 43.6%

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

Alternative 15: 49.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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 = y / (z * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

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 49.8

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

4. Applied rewrites 49.8%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))