Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.0% → 96.2%

Time: 15.0s

Alternatives: 14

Speedup: 2.6×

• 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

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((cosh(x) * y) / x) / z
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(Float64(Float64(cosh(x) * 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[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cosh-lowering-cosh.f64 98.4

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

4. Applied egg-rr 98.4%

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

Alternative 2: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot \left(x \cdot x\right)}{z} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 INFINITY)
     (/ t_0 z)
     (* y (* (/ (* x (* x x)) z) 0.041666666666666664)))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / z;
	} else {
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 / z
	else:
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y * Float64(Float64(Float64(x * Float64(x * x)) / z) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 / z;
	else
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 / z), $MachinePrecision], N[(y * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 0.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 0.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{{x}^{3} \cdot y}{z} \cdot \frac{1}{24}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot {x}^{3}}}{z} \cdot \frac{1}{24} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\left(y \cdot \frac{{x}^{3}}{z}\right)} \cdot \frac{1}{24} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{{x}^{3}}{z} \cdot \frac{1}{24}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{{x}^{3}}{z} \cdot \frac{1}{24}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{{x}^{3}}{z} \cdot \frac{1}{24}\right)} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{{x}^{3}}{z}} \cdot \frac{1}{24}\right) \]

       8. cube-mult N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

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

Alternative 3: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+101}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (cosh x) (/ y x)) 1e+101)
   (/ (/ y x) z)
   (/
    (/
     (*
      y
      (fma
       (fma x x 0.0)
       (fma
        x
        (* x (fma (fma x x 0.0) 0.001388888888888889 0.041666666666666664))
        0.5)
       1.0))
     z)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((cosh(x) * (y / x)) <= 1e+101) {
		tmp = (y / x) / z;
	} else {
		tmp = ((y * fma(fma(x, x, 0.0), fma(x, (x * fma(fma(x, x, 0.0), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)) / z) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y / x)) <= 1e+101)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(Float64(y * fma(fma(x, x, 0.0), fma(x, Float64(x * fma(fma(x, x, 0.0), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)) / z) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], 1e+101], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y * N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x + 0.0), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999998e100

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 64.5

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

   5. Simplified 64.5%

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

   if 9.9999999999999998e100 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 79.2%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cosh-lowering-cosh.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 89.6

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

   7. Simplified 89.6%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

   10. Simplified 69.0%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. /-lowering-/.f64 N/A

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

   12. Applied egg-rr 91.3%

       \[\leadsto \color{blue}{\frac{\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.4e-14)
   (/ (/ y x) z)
   (if (<= x 2.8e+69)
     (* y (/ (cosh x) (* x z)))
     (/
      (* y (/ (fma (* x x) (fma x (* x 0.041666666666666664) 0.5) 1.0) x))
      z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.4e-14) {
		tmp = (y / x) / z;
	} else if (x <= 2.8e+69) {
		tmp = y * (cosh(x) / (x * z));
	} else {
		tmp = (y * (fma((x * x), fma(x, (x * 0.041666666666666664), 0.5), 1.0) / x)) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.4e-14)
		tmp = Float64(Float64(y / x) / z);
	elseif (x <= 2.8e+69)
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(Float64(y * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), 0.5), 1.0) / x)) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.4e-14], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 2.8e+69], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.3999999999999997e-14

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 65.9

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

   5. Simplified 65.9%

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

   if 5.3999999999999997e-14 < x < 2.79999999999999982e69

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. cosh-lowering-cosh.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 92.9

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

   4. Applied egg-rr 92.9%

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

   if 2.79999999999999982e69 < x

   1. Initial program 82.1%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

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

Alternative 5: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.6e+42)
   (/ (cosh x) (* z (/ x y)))
   (/
    (/
     (*
      y
      (fma
       (* x x)
       (fma
        (* x x)
        (fma x (* x 0.001388888888888889) 0.041666666666666664)
        0.5)
       1.0))
     x)
    z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.6e+42) {
		tmp = cosh(x) / (z * (x / y));
	} else {
		tmp = ((y * fma((x * x), fma((x * x), fma(x, (x * 0.001388888888888889), 0.041666666666666664), 0.5), 1.0)) / x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.6e+42)
		tmp = Float64(cosh(x) / Float64(z * Float64(x / y)));
	else
		tmp = Float64(Float64(Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664), 0.5), 1.0)) / x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.6e+42], N[(N[Cosh[x], $MachinePrecision] / N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e42

   1. Initial program 91.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. cosh-lowering-cosh.f64 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 85.6

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

   4. Applied egg-rr 85.6%

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

   if 4.6e42 < x

   1. Initial program 83.3%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (/
   (*
    y
    (fma
     (* x x)
     (fma (* x x) (fma x (* x 0.001388888888888889) 0.041666666666666664) 0.5)
     1.0))
   x)
  z))

C

double code(double x, double y, double z) {
	return ((y * fma((x * x), fma((x * x), fma(x, (x * 0.001388888888888889), 0.041666666666666664), 0.5), 1.0)) / x) / z;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664), 0.5), 1.0)) / x) / z)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cosh-lowering-cosh.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 92.0

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

7. Simplified 92.0%

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

8. Final simplification 92.0%

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

Alternative 7: 89.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(0.001388888888888889 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)\right), y\right)}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (/ (fma (* x x) (* x (* x (* 0.001388888888888889 (* y (* x x))))) y) x)
  z))

C

double code(double x, double y, double z) {
	return (fma((x * x), (x * (x * (0.001388888888888889 * (y * (x * x))))), y) / x) / z;
}

Julia

function code(x, y, z)
	return Float64(Float64(fma(Float64(x * x), Float64(x * Float64(x * Float64(0.001388888888888889 * Float64(y * Float64(x * x))))), y) / x) / z)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(0.001388888888888889 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cosh-lowering-cosh.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

7. Simplified 90.9%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 90.5

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

10. Simplified 90.5%

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

11. Final simplification 90.5%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(0.001388888888888889 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)\right), y\right)}{x}}{z} \]
12. Add Preprocessing

Alternative 8: 88.4% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (/ (* y (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0)) x) z))

C

double code(double x, double y, double z) {
	return ((y * fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0)) / x) / z;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0)) / x) / z)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cosh-lowering-cosh.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 92.0

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

7. Simplified 92.0%

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

8. Taylor expanded in x around 0

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

10. Simplified 89.7%

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

11. Final simplification 89.7%

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

Alternative 9: 87.9% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (* y (/ (fma (* x x) (fma x (* x 0.041666666666666664) 0.5) 1.0) x)) z))

C

double code(double x, double y, double z) {
	return (y * (fma((x * x), fma(x, (x * 0.041666666666666664), 0.5), 1.0) / x)) / z;
}

Julia

function code(x, y, z)
	return Float64(Float64(y * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), 0.5), 1.0) / x)) / z)
end

Wolfram

code[x_, y_, z_] := N[(N[(y * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. cosh-lowering-cosh.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

7. Simplified 89.7%

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

Alternative 10: 67.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00028)
   (/ (/ y x) z)
   (/ (* y (* (* x (* x x)) 0.041666666666666664)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = (y / x) / z;
	} else {
		tmp = (y * ((x * (x * x)) * 0.041666666666666664)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.00028d0) then
        tmp = (y / x) / z
    else
        tmp = (y * ((x * (x * x)) * 0.041666666666666664d0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = (y / x) / z;
	} else {
		tmp = (y * ((x * (x * x)) * 0.041666666666666664)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.00028:
		tmp = (y / x) / z
	else:
		tmp = (y * ((x * (x * x)) * 0.041666666666666664)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00028)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(y * Float64(Float64(x * Float64(x * x)) * 0.041666666666666664)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.00028)
		tmp = (y / x) / z;
	else
		tmp = (y * ((x * (x * x)) * 0.041666666666666664)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00028], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998e-4

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 66.0

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

   5. Simplified 66.0%

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

   if 2.7999999999999998e-4 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 69.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 69.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.6

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 79.6

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

   11. Simplified 79.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot \left(x \cdot x\right)}{z} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00028)
   (/ (/ y x) z)
   (* y (* (/ (* x (* x x)) z) 0.041666666666666664))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = (y / x) / z;
	} else {
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.00028d0) then
        tmp = (y / x) / z
    else
        tmp = y * (((x * (x * x)) / z) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = (y / x) / z;
	} else {
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.00028:
		tmp = (y / x) / z
	else:
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00028)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y * Float64(Float64(Float64(x * Float64(x * x)) / z) * 0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.00028)
		tmp = (y / x) / z;
	else
		tmp = y * (((x * (x * x)) / z) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00028], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998e-4

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 66.0

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

   5. Simplified 66.0%

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

   if 2.7999999999999998e-4 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 69.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 69.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.6

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{{x}^{3} \cdot y}{z} \cdot \frac{1}{24}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot {x}^{3}}}{z} \cdot \frac{1}{24} \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\left(y \cdot \frac{{x}^{3}}{z}\right)} \cdot \frac{1}{24} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{{x}^{3}}{z} \cdot \frac{1}{24}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{{x}^{3}}{z} \cdot \frac{1}{24}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{{x}^{3}}{z} \cdot \frac{1}{24}\right)} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{{x}^{3}}{z}} \cdot \frac{1}{24}\right) \]

       8. cube-mult N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 77.8

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

   11. Simplified 77.8%

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

Alternative 12: 58.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00028) (/ (/ y x) z) (/ (* y (* x 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = (y / x) / z;
	} else {
		tmp = (y * (x * 0.5)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.00028d0) then
        tmp = (y / x) / z
    else
        tmp = (y * (x * 0.5d0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = (y / x) / z;
	} else {
		tmp = (y * (x * 0.5)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.00028:
		tmp = (y / x) / z
	else:
		tmp = (y * (x * 0.5)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00028)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(y * Float64(x * 0.5)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.00028)
		tmp = (y / x) / z;
	else
		tmp = (y * (x * 0.5)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00028], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998e-4

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 66.0

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

   5. Simplified 66.0%

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

   if 2.7999999999999998e-4 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. *-rgt-identity N/A

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

      19. *-commutative N/A

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

      20. accelerator-lowering-fma.f64 N/A

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

      21. /-lowering-/.f64 57.6

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

   5. Simplified 57.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 57.6

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

   8. Simplified 57.6%

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

Alternative 13: 58.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00028) (/ y (* x z)) (/ (* y (* x 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = y / (x * z);
	} else {
		tmp = (y * (x * 0.5)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 0.00028d0) then
        tmp = y / (x * z)
    else
        tmp = (y * (x * 0.5d0)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00028) {
		tmp = y / (x * z);
	} else {
		tmp = (y * (x * 0.5)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.00028:
		tmp = y / (x * z)
	else:
		tmp = (y * (x * 0.5)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00028)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y * Float64(x * 0.5)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.00028)
		tmp = y / (x * z);
	else
		tmp = (y * (x * 0.5)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00028], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998e-4

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 63.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 63.2

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

   7. Applied egg-rr 63.2%

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

   if 2.7999999999999998e-4 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. distribute-lft-out N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. *-rgt-identity N/A

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

      19. *-commutative N/A

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

      20. accelerator-lowering-fma.f64 N/A

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

      21. /-lowering-/.f64 57.6

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

   5. Simplified 57.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 57.6

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

   8. Simplified 57.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y / (x * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-rgt-identity N/A

      \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

   3. accelerator-lowering-fma.f64 52.1

      \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

5. Simplified 52.1%

   \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 52.1

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

7. Applied egg-rr 52.1%

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

8. Final simplification 52.1%

   \[\leadsto \frac{y}{x \cdot z} \]
9. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

  (/ (* (cosh x) (/ y x)) z))