Average Error: 13.7 → 1.5
Time: 15.9s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -2.84628236833302262e159

    1. Initial program 39.7

      \[\]

    2. Simplified35.3

      \[\leadsto \]
    3. Using strategy rm

    4. Applied associate-*r/35.3

      \[\leadsto \]

    5. Simplified35.3

      \[\leadsto \]

    6. Taylor expanded around -inf 3.8

      \[\leadsto \]

    7. Simplified3.8

      \[\leadsto \]

    if -2.84628236833302262e159 < F < 1.3316745337714515e154

    1. Initial program 2.8

      \[\]

    2. Simplified0.7

      \[\leadsto \]
    3. Using strategy rm

    4. Applied associate-*r/0.6

      \[\leadsto \]

    if 1.3316745337714515e154 < F

    1. Initial program 41.8

      \[\]

    2. Simplified36.6

      \[\leadsto \]
    3. Using strategy rm

    4. Applied associate-*r/36.6

      \[\leadsto \]

    5. Simplified36.6

      \[\leadsto \]

    6. Taylor expanded around inf 4.0

      \[\leadsto \]

    7. Simplified4.0

      \[\leadsto \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \]

Reproduce

herbie shell --seed 2020179 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (neg (/ 1.0 2.0))))))