Average Error: 0.0 → 0.6
Time: 3.8s
Precision: 64
\[\sin x \cdot \frac{\sinh y}{y}\]
↓
\[\sin x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\]
\sin x \cdot \frac{\sinh y}{y}
↓
\sin x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)
double code(double x, double y) {
	return ((double) (((double) sin(x)) * ((double) (((double) sinh(y)) / y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) sin(x)) * ((double) fma(0.16666666666666666, ((double) pow(y, 2.0)), ((double) fma(0.008333333333333333, ((double) pow(y, 4.0)), 1.0))))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.6

    \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \left(\frac{1}{120} \cdot {y}^{4} + 1\right)\right)}\]

  3. Simplified0.6

    \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))