Linear.Quaternion:$csin from linear-1.19.1.3

Average Error: 0.02% → 0.02%

Time: 8.3s

Precision: binary64

Cost: 13120

Math

\[\cos x \cdot \frac{\sinh y}{y} \]

↓

\[\frac{\cos x}{\frac{y}{\sinh y}} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

↓

(FPCore (x y) :precision binary64 (/ (cos x) (/ y (sinh y))))

C

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

↓

double code(double x, double y) {
	return cos(x) / (y / sinh(y));
}

Fortran

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) / (y / sinh(y))
end function

Java

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

↓

public static double code(double x, double y) {
	return Math.cos(x) / (y / Math.sinh(y));
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

↓

def code(x, y):
	return math.cos(x) / (y / math.sinh(y))

Julia

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

↓

function code(x, y)
	return Float64(cos(x) / Float64(y / sinh(y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = cos(x) / (y / sinh(y));
end

Wolfram

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

↓

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

   \[\cos x \cdot \frac{\sinh y}{y} \]

2. Applied egg-rr 0.02

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

3. Final simplification 0.02

   \[\leadsto \frac{\cos x}{\frac{y}{\sinh y}} \]

Alternatives

Alternative 1
Error	0.92%
Cost	13636

\[\begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 1.0000002:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.28%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\cos x \leq 1:\\ \;\;\;\;\frac{\cos x}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 3
Error	1.79%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\cos x \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 4
Error	0.02%
Cost	13120

\[\cos x \cdot \frac{\sinh y}{y} \]

Alternative 5
Error	0.97%
Cost	7360

\[\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333 + 0.16666666666666666\right)\right) \]

Alternative 6
Error	1.79%
Cost	6464

\[\cos x \]

Alternative 7
Error	45.71%
Cost	448

\[1 + \left(y \cdot y\right) \cdot 0.16666666666666666 \]

Alternative 8
Error	46.02%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))