Linear.Quaternion:$ccos from linear-1.19.1.3

Average Error: 0.0 → 0.2

Time: 7.2s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = sinh(y) / (y / sin(x))
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.sinh(y) / (y / Math.sin(x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.2

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

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.0
Cost	6464

\[\sin x \]

Alternative 2
Error	31.6
Cost	576

\[\frac{-1}{x \cdot -0.16666666666666666 + \frac{-1}{x}} \]

Alternative 3
Error	31.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))