Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 22.27% → 0.16%

Time: 10.0s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	22.27%
Target	0.31%
Herbie	0.16%

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

Derivation

1. Initial program 22.27

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

2. Simplified 0.16

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

   [Start] 22.27	\[ \frac{\sin x \cdot \sinh y}{x} \]
   associate-*l/ [<=] 0.16	\[ \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

3. Final simplification 0.16

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

Alternatives

Alternative 1
Error	2.07%
Cost	6720

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

Alternative 2
Error	1.98%
Cost	6720

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

Alternative 3
Error	25.39%
Cost	704

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

Alternative 4
Error	27.39%
Cost	320

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

Alternative 5
Error	48.32%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023103 
(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))