Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 2.0s

Precision: binary64

Math

\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \left(-0.25 \cdot \mathsf{fma}\left(v, v, -1\right)\right)\right)\right)\right) \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))

↓

(FPCore (v)
 :precision binary64
 (expm1
  (log1p
   (* (sqrt 2.0) (* (sqrt (fma v (* v -3.0) 1.0)) (* -0.25 (fma v v -1.0)))))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt(1.0 - (3.0 * (v * v)))) * (1.0 - (v * v));
}

↓

double code(double v) {
	return expm1(log1p(sqrt(2.0) * (sqrt(fma(v, (v * -3.0), 1.0)) * (-0.25 * fma(v, v, -1.0)))));
}

Error

Bits error versus v

Derivation

1. Initial program 0.0

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \left(-0.25 \cdot \mathsf{fma}\left(v, v, -1\right)\right)\right)} \]

3. Applied expm1-log1p-u_binary64 0.0

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \left(-0.25 \cdot \mathsf{fma}\left(v, v, -1\right)\right)\right)\right)\right)} \]

4. Final simplification 0.0

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \left(-0.25 \cdot \mathsf{fma}\left(v, v, -1\right)\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022103 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))