Linear.Quaternion:$ccos from linear-1.19.1.3

Average Accuracy: 99.9% → 99.9%

Time: 12.9s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \sin x \cdot \frac{\sinh y}{y} \]
   associate-*r/ [=>] 78.1	\[ \color{blue}{\frac{\sin x \cdot \sinh y}{y}} \]
   associate-/l* [=>] 99.9	\[ \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13120

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

Alternative 2
Accuracy	98.9%
Cost	7360

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

Alternative 3
Accuracy	98.7%
Cost	6976

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

Alternative 4
Accuracy	98.2%
Cost	6464

\[\sin x \]

Alternative 5
Accuracy	50.6%
Cost	576

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

Alternative 6
Accuracy	50.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))