Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 14.3 → 0.1

Time: 15.4s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, y)
	tmp = sinh(y) * (sin(x) / 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[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Target

Original	14.3
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 14.3

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

2. Simplified 0.1

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

   [Start] 14.3	\[ \frac{\sin x \cdot \sinh y}{x} \]
   rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\sinh y \cdot \frac{\sin x}{x}} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	13120

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

Alternative 2
Error	1.3
Cost	6720

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

Alternative 3
Error	16.4
Cost	832

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

Alternative 4
Error	16.5
Cost	704

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

Alternative 5
Error	17.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(0.16666666666666666 \cdot x\right) \cdot x}\\ \end{array} \]

Alternative 6
Error	18.0
Cost	576

\[\left(x + x\right) \cdot \left(y \cdot \frac{0.5}{x}\right) \]

Alternative 7
Error	17.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y - -1\right) + -1\\ \end{array} \]

Alternative 8
Error	17.9
Cost	320

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

Alternative 9
Error	30.9
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023074 
(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))