Average Error: 0.2 → 0.2
Time: 6.3s
Precision: binary64
\[\cosh x \cdot \frac{\sin y}{y} \]
↓
\[\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, e^{x}, \frac{0.5}{e^{x}}\right) \]
(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))
↓
(FPCore (x y)
 :precision binary64
 (* (/ (sin y) y) (fma 0.5 (exp x) (/ 0.5 (exp x)))))
double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}
↓
double code(double x, double y) {
	return (sin(y) / y) * fma(0.5, exp(x), (0.5 / exp(x)));
}

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end
↓
function code(x, y)
	return Float64(Float64(sin(y) / y) * fma(0.5, exp(x), Float64(0.5 / exp(x))))
end

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(0.5 * N[Exp[x], $MachinePrecision] + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, e^{x}, \frac{0.5}{e^{x}}\right)

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[\frac{\cosh x \cdot \sin y}{y} \]

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded in x around inf 0.2

    \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y \cdot \left(\frac{1}{e^{x}} + e^{x}\right)}{y}} \]

  3. Simplified0.2

    \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, e^{x}, \frac{0.5}{e^{x}}\right)} \]

  4. Final simplification0.2

    \[\leadsto \frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, e^{x}, \frac{0.5}{e^{x}}\right) \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))