Linear.Quaternion:$ccosh from linear-1.19.1.3

?

Percentage Accurate: 89.3% → 99.9%
Time: 8.3s
Precision: binary64
Cost: 13120

?

\[\frac{\sin x \cdot \sinh y}{x} \]
↓
\[\frac{\sinh y}{\frac{x}{\sin x}} \]
(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))
↓
(FPCore (x y) :precision binary64 (/ (sinh y) (/ x (sin x))))
double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}
↓
double code(double x, double y) {
	return sinh(y) / (x / sin(x));
}

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

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

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

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

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

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

\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\sinh y}{\frac{x}{\sin x}}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 13 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original89.3%
Target99.9%
Herbie99.9%
\[\sin x \cdot \frac{\sinh y}{x} \]

Derivation?

  1. Initial program 87.7%

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

  2. Simplified99.9%

    \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
    Step-by-step derivation

    [Start]87.7%

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

    associate-*r/ [<=]99.9%

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

  3. Applied egg-rr99.9%

    \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{x}} \]
    Step-by-step derivation

    [Start]99.9%

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

    add-log-exp [=>]65.4%

    \[ \color{blue}{\log \left(e^{\sin x \cdot \frac{\sinh y}{x}}\right)} \]

    *-un-lft-identity [=>]65.4%

    \[ \log \color{blue}{\left(1 \cdot e^{\sin x \cdot \frac{\sinh y}{x}}\right)} \]

    log-prod [=>]65.4%

    \[ \color{blue}{\log 1 + \log \left(e^{\sin x \cdot \frac{\sinh y}{x}}\right)} \]

    metadata-eval [=>]65.4%

    \[ \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{x}}\right) \]

    add-log-exp [<=]99.9%

    \[ 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

  4. Simplified99.9%

    \[\leadsto \color{blue}{\frac{\sinh y}{\frac{x}{\sin x}}} \]
    Step-by-step derivation

    [Start]99.9%

    \[ 0 + \sin x \cdot \frac{\sinh y}{x} \]

    +-lft-identity [=>]99.9%

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

    *-commutative [=>]99.9%

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

    associate-/r/ [<=]99.9%

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

  5. Final simplification99.9%

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

Alternatives

Alternative 1
Accuracy99.9%
Cost13120
\[\frac{\sinh y}{\frac{x}{\sin x}} \]
Alternative 2
Accuracy86.2%
Cost19784
\[\begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
Alternative 3
Accuracy86.3%
Cost19784
\[\begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
Alternative 4
Accuracy86.3%
Cost19784
\[\begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
Alternative 5
Accuracy72.2%
Cost19528
\[\begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-156}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
Alternative 6
Accuracy99.9%
Cost13120
\[\sin x \cdot \frac{\sinh y}{x} \]
Alternative 7
Accuracy99.9%
Cost13120
\[\frac{\sin x}{x} \cdot \sinh y \]
Alternative 8
Accuracy87.1%
Cost7372
\[\begin{array}{l} t_0 := \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{if}\;y \leq -0.0017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+75}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Accuracy52.6%
Cost6724
\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \mathbf{elif}\;x \leq 68000000000000:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y}\\ \end{array} \]
Alternative 10
Accuracy53.1%
Cost649
\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \lor \neg \left(x \leq 68000000000000\right):\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]
Alternative 11
Accuracy48.5%
Cost648
\[\begin{array}{l} \mathbf{if}\;x \leq 1.58 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]
Alternative 12
Accuracy49.3%
Cost320
\[x \cdot \frac{y}{x} \]
Alternative 13
Accuracy27.1%
Cost64
\[y \]

Reproduce?

herbie shell --seed 2023229 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))