Average Error: 39.2 → 0.1
Time: 55.8s
Precision: 64
Internal Precision: 1408
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left({x}^{3} + 1\right) - \left(\log \left({\left(x \cdot x\right)}^{3} + {\left(1 - x\right)}^{3}\right) - \log \left(\left(\left(1 - x\right) - x \cdot x\right) \cdot \left(1 - x\right) + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \le -0.0024829857670106417:\\ \;\;\;\;\log \left({x}^{3} + 1\right) - \log \left(x \cdot x + \left(1 - x\right)\right)\\ \mathbf{if}\;\log \left({x}^{3} + 1\right) - \left(\log \left({\left(x \cdot x\right)}^{3} + {\left(1 - x\right)}^{3}\right) - \log \left(\left(\left(1 - x\right) - x \cdot x\right) \cdot \left(1 - x\right) + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \le 0.0015651223709737463:\\ \;\;\;\;x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original39.2
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (log (+ (pow x 3) 1)) (- (log (+ (pow (* x x) 3) (pow (- 1 x) 3))) (log (+ (* (- (- 1 x) (* x x)) (- 1 x)) (* (* x x) (* x x)))))) < -0.0024829857670106417

    1. Initial program 0.3

      \[\log \left(1 + x\right)\]
    2. Using strategy rm

    3. Applied flip3-+0.4

      \[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}\right)}\]

    4. Applied log-div1.1

      \[\leadsto \color{blue}{\log \left({1}^{3} + {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}\]

    5. Applied simplify1.1

      \[\leadsto \color{blue}{\log \left({x}^{3} + 1\right)} - \log \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\]

    6. Applied simplify1.1

      \[\leadsto \log \left({x}^{3} + 1\right) - \color{blue}{\log \left(x \cdot x + \left(1 - x\right)\right)}\]

    if -0.0024829857670106417 < (- (log (+ (pow x 3) 1)) (- (log (+ (pow (* x x) 3) (pow (- 1 x) 3))) (log (+ (* (- (- 1 x) (* x x)) (- 1 x)) (* (* x x) (* x x)))))) < 0.0015651223709737463

    1. Initial program 59.1

      \[\log \left(1 + x\right)\]

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}}\]

    3. Applied simplify0.1

      \[\leadsto \color{blue}{x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}\]

    if 0.0015651223709737463 < (- (log (+ (pow x 3) 1)) (- (log (+ (pow (* x x) 3) (pow (- 1 x) 3))) (log (+ (* (- (- 1 x) (* x x)) (- 1 x)) (* (* x x) (* x x))))))

    1. Initial program 0.0

      \[\log \left(1 + x\right)\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 55.8s)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))