Linear.Quaternion:$cexp from linear-1.19.1.3

Average Accuracy: 99.8% → 99.8%

Time: 8.9s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ x \cdot \frac{\sin y}{y} \]
   clear-num [=>] 99.7	\[ x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}} \]
   un-div-inv [=>] 99.8	\[ \color{blue}{\frac{x}{\frac{y}{\sin y}}} \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	6720

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

Alternative 2
Accuracy	63.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 680000000000:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	63.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	63.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;6 \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	62.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+71} \lor \neg \left(y \leq 0.02\right):\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	63.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+27} \lor \neg \left(y \leq 1.8 \cdot 10^{-6}\right):\\ \;\;\;\;\left(x + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	63.6%
Cost	576

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

Alternative 8
Accuracy	51.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))