Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 22.54% → 0.19%

Time: 8.7s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

Java

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));
}

Python

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

↓

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

Julia

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

MATLAB

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

↓

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

Wolfram

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]

Target

Original	22.54%
Target	0.39%
Herbie	0.19%

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

Derivation

1. Initial program 22.54

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

2. Simplified 0.19

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

   [Start] 22.54	\[ \frac{\sin x \cdot \sinh y}{x} \]
   *-commutative [=>] 22.54	\[ \frac{\color{blue}{\sinh y \cdot \sin x}}{x} \]
   associate-/l* [=>] 0.19	\[ \color{blue}{\frac{\sinh y}{\frac{x}{\sin x}}} \]

3. Final simplification 0.19

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

Alternatives

Alternative 1
Error	0.19%
Cost	13120

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

Alternative 2
Error	2.03%
Cost	6720

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

Alternative 3
Error	1.94%
Cost	6720

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

Alternative 4
Error	1.94%
Cost	6720

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

Alternative 5
Error	27.01%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 1\right) + -1\\ \end{array} \]

Alternative 6
Error	27.76%
Cost	320

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

Alternative 7
Error	26.49%
Cost	320

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

Alternative 8
Error	48.56%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023090 
(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))