Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Accuracy: 78.1% → 99.8%

Time: 10.3s

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	78.1%
Target	99.7%
Herbie	99.8%

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

Derivation

1. Initial program 78.1%

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

2. Simplified 99.8%

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

   [Start] 78.1	\[ \frac{\sin x \cdot \sinh y}{x} \]
   associate-*l/ [<=] 99.8	\[ \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	13120

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

Alternative 2
Accuracy	99.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-8} \lor \neg \left(x \leq 4.7 \cdot 10^{-8}\right):\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	6720

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

Alternative 4
Accuracy	73.5%
Cost	704

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

Alternative 5
Accuracy	73.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+15}:\\ \;\;\;\;\left(y + 1\right) + -1\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	73.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	73.3%
Cost	320

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

Alternative 8
Accuracy	51.2%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023146 
(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))