Average Error: 39.1 → 0.3
Time: 54.0s
Precision: 64
Internal Precision: 1344
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(\left(\log \left(1 + {x}^{3}\right) - \log \left({\left(x \cdot x\right)}^{3} + {\left(1 - x\right)}^{3}\right)\right) + \log \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 - x\right) \cdot \left(1 - x\right) - \left(x \cdot x\right) \cdot \left(1 - x\right)\right)\right)\right)}^{3}} \le -0.003366078258316278:\\ \;\;\;\;\log \left(1 + x\right)\\ \mathbf{if}\;\sqrt[3]{{\left(\left(\log \left(1 + {x}^{3}\right) - \log \left({\left(x \cdot x\right)}^{3} + {\left(1 - x\right)}^{3}\right)\right) + \log \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 - x\right) \cdot \left(1 - x\right) - \left(x \cdot x\right) \cdot \left(1 - x\right)\right)\right)\right)}^{3}} \le 5.434804327505188:\\ \;\;\;\;x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cbrt (pow (+ (- (log (+ 1 (pow x 3))) (log (+ (pow (* x x) 3) (pow (- 1 x) 3)))) (log (+ (* (* x x) (* x x)) (- (* (- 1 x) (- 1 x)) (* (* x x) (- 1 x)))))) 3)) < -0.003366078258316278 or 5.434804327505188 < (cbrt (pow (+ (- (log (+ 1 (pow x 3))) (log (+ (pow (* x x) 3) (pow (- 1 x) 3)))) (log (+ (* (* x x) (* x x)) (- (* (- 1 x) (- 1 x)) (* (* x x) (- 1 x)))))) 3))

    1. Initial program 0.0

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

    if -0.003366078258316278 < (cbrt (pow (+ (- (log (+ 1 (pow x 3))) (log (+ (pow (* x x) 3) (pow (- 1 x) 3)))) (log (+ (* (* x x) (* x x)) (- (* (- 1 x) (- 1 x)) (* (* x x) (- 1 x)))))) 3)) < 5.434804327505188

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.4

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

    3. Applied simplify0.4

      \[\leadsto \color{blue}{x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 54.0s)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))