Quotient of products

Percentage Accurate: 86.1% → 96.8%
Time: 4.0s
Alternatives: 7
Speedup: TODO×

Specification

?
\[\frac{a1 \cdot a2}{b1 \cdot b2} \]

Your Program's Arguments

Results

Enter valid numbers for all inputs

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 7 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Target

Original86.1%
Target86.5%
Herbie96.8%
\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

Alternative 1?

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-312}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or -9.9999999999847e-313 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.99999999999999992e-291

    1. Initial program 80.2%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac96.6%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified96.6%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
    4. Step-by-step derivation

      1. frac-times80.2%

        \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b1 \cdot b2}} \]

      2. *-commutative80.2%

        \[\leadsto \frac{a1 \cdot a2}{\color{blue}{b2 \cdot b1}} \]

      3. frac-times94.3%

        \[\leadsto \color{blue}{\frac{a1}{b2} \cdot \frac{a2}{b1}} \]

      4. clear-num93.2%

        \[\leadsto \frac{a1}{b2} \cdot \color{blue}{\frac{1}{\frac{b1}{a2}}} \]

      5. un-div-inv93.2%

        \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{\frac{b1}{a2}}} \]

    5. Applied egg-rr93.2%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2}}{\frac{b1}{a2}}} \]

    if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.9999999999847e-313 or 1.99999999999999992e-291 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1e308

    1. Initial program 99.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

    if 1e308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

    1. Initial program 68.5%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac96.8%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified96.8%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
    4. Step-by-step derivation

      1. associate-*r/99.8%

        \[\leadsto \color{blue}{\frac{\frac{a1}{b1} \cdot a2}{b2}} \]

    5. Applied egg-rr99.8%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1} \cdot a2}{b2}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification96.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \end{array} \]

Alternative 2?

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

    1. Initial program 84.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. associate-/l*92.4%

        \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]

      2. *-commutative92.4%

        \[\leadsto \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]

      3. associate-/l*97.3%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

    3. Simplified97.3%

      \[\leadsto \color{blue}{\frac{a1}{\frac{b2}{\frac{a2}{b1}}}} \]
    4. Step-by-step derivation

      1. associate-/r/97.3%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    5. Applied egg-rr97.3%

      \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2e-276 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1e308

    1. Initial program 99.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

    if -2e-276 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

    1. Initial program 77.7%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac95.2%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified95.2%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
    4. Step-by-step derivation

      1. clear-num95.2%

        \[\leadsto \color{blue}{\frac{1}{\frac{b1}{a1}}} \cdot \frac{a2}{b2} \]

      2. frac-times93.8%

        \[\leadsto \color{blue}{\frac{1 \cdot a2}{\frac{b1}{a1} \cdot b2}} \]

      3. *-un-lft-identity93.8%

        \[\leadsto \frac{\color{blue}{a2}}{\frac{b1}{a1} \cdot b2} \]

    5. Applied egg-rr93.8%

      \[\leadsto \color{blue}{\frac{a2}{\frac{b1}{a1} \cdot b2}} \]

    if 1e308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

    1. Initial program 68.5%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac96.8%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified96.8%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification97.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 3?

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

    1. Initial program 84.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. associate-/l*92.4%

        \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]

      2. *-commutative92.4%

        \[\leadsto \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]

      3. associate-/l*97.3%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

    3. Simplified97.3%

      \[\leadsto \color{blue}{\frac{a1}{\frac{b2}{\frac{a2}{b1}}}} \]
    4. Step-by-step derivation

      1. associate-/r/97.3%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    5. Applied egg-rr97.3%

      \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2e-276 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1e308

    1. Initial program 99.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

    if -2e-276 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

    1. Initial program 77.7%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac95.2%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified95.2%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
    4. Step-by-step derivation

      1. clear-num95.2%

        \[\leadsto \color{blue}{\frac{1}{\frac{b1}{a1}}} \cdot \frac{a2}{b2} \]

      2. frac-times93.8%

        \[\leadsto \color{blue}{\frac{1 \cdot a2}{\frac{b1}{a1} \cdot b2}} \]

      3. *-un-lft-identity93.8%

        \[\leadsto \frac{\color{blue}{a2}}{\frac{b1}{a1} \cdot b2} \]

    5. Applied egg-rr93.8%

      \[\leadsto \color{blue}{\frac{a2}{\frac{b1}{a1} \cdot b2}} \]

    if 1e308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

    1. Initial program 68.5%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. associate-/l*75.2%

        \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]

      2. *-commutative75.2%

        \[\leadsto \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]

      3. associate-/l*91.6%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

    3. Simplified91.6%

      \[\leadsto \color{blue}{\frac{a1}{\frac{b2}{\frac{a2}{b1}}}} \]
    4. Step-by-step derivation

      1. add-sqr-sqrt48.2%

        \[\leadsto \frac{\color{blue}{\sqrt{a1} \cdot \sqrt{a1}}}{\frac{b2}{\frac{a2}{b1}}} \]

      2. div-inv48.2%

        \[\leadsto \frac{\sqrt{a1} \cdot \sqrt{a1}}{\color{blue}{b2 \cdot \frac{1}{\frac{a2}{b1}}}} \]

      3. times-frac48.2%

        \[\leadsto \color{blue}{\frac{\sqrt{a1}}{b2} \cdot \frac{\sqrt{a1}}{\frac{1}{\frac{a2}{b1}}}} \]

      4. clear-num48.2%

        \[\leadsto \frac{\sqrt{a1}}{b2} \cdot \frac{\sqrt{a1}}{\color{blue}{\frac{b1}{a2}}} \]

    5. Applied egg-rr48.2%

      \[\leadsto \color{blue}{\frac{\sqrt{a1}}{b2} \cdot \frac{\sqrt{a1}}{\frac{b1}{a2}}} \]
    6. Step-by-step derivation

      1. associate-*l/48.3%

        \[\leadsto \color{blue}{\frac{\sqrt{a1} \cdot \frac{\sqrt{a1}}{\frac{b1}{a2}}}{b2}} \]

      2. div-inv48.3%

        \[\leadsto \frac{\sqrt{a1} \cdot \color{blue}{\left(\sqrt{a1} \cdot \frac{1}{\frac{b1}{a2}}\right)}}{b2} \]

      3. clear-num48.3%

        \[\leadsto \frac{\sqrt{a1} \cdot \left(\sqrt{a1} \cdot \color{blue}{\frac{a2}{b1}}\right)}{b2} \]

      4. associate-*r*48.3%

        \[\leadsto \frac{\color{blue}{\left(\sqrt{a1} \cdot \sqrt{a1}\right) \cdot \frac{a2}{b1}}}{b2} \]

      5. add-sqr-sqrt99.8%

        \[\leadsto \frac{\color{blue}{a1} \cdot \frac{a2}{b1}}{b2} \]

    7. Applied egg-rr99.8%

      \[\leadsto \color{blue}{\frac{a1 \cdot \frac{a2}{b1}}{b2}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification97.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array} \]

Alternative 4?

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

    1. Initial program 84.8%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. associate-/l*92.4%

        \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]

      2. *-commutative92.4%

        \[\leadsto \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]

      3. associate-/l*97.3%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

    3. Simplified97.3%

      \[\leadsto \color{blue}{\frac{a1}{\frac{b2}{\frac{a2}{b1}}}} \]
    4. Step-by-step derivation

      1. associate-/r/97.3%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    5. Applied egg-rr97.3%

      \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2e-276 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1e308

    1. Initial program 99.3%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

    if -2e-276 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

    1. Initial program 77.7%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac95.2%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified95.2%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
    4. Step-by-step derivation

      1. clear-num95.2%

        \[\leadsto \color{blue}{\frac{1}{\frac{b1}{a1}}} \cdot \frac{a2}{b2} \]

      2. frac-times93.8%

        \[\leadsto \color{blue}{\frac{1 \cdot a2}{\frac{b1}{a1} \cdot b2}} \]

      3. *-un-lft-identity93.8%

        \[\leadsto \frac{\color{blue}{a2}}{\frac{b1}{a1} \cdot b2} \]

    5. Applied egg-rr93.8%

      \[\leadsto \color{blue}{\frac{a2}{\frac{b1}{a1} \cdot b2}} \]

    if 1e308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

    1. Initial program 68.5%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac96.8%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified96.8%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
    4. Step-by-step derivation

      1. associate-*r/99.8%

        \[\leadsto \color{blue}{\frac{\frac{a1}{b1} \cdot a2}{b2}} \]

    5. Applied egg-rr99.8%

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1} \cdot a2}{b2}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification97.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2 \cdot 10^{-276}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b2 \cdot \frac{b1}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 10^{+308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \end{array} \]

Alternative 5?

\[\begin{array}{l} \mathbf{if}\;b1 \leq -3 \cdot 10^{-27}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if b1 < -3.0000000000000001e-27

    1. Initial program 90.4%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. associate-/l*93.2%

        \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}} \]

      2. *-commutative93.2%

        \[\leadsto \frac{a1}{\frac{\color{blue}{b2 \cdot b1}}{a2}} \]

      3. associate-/l*89.2%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]

    3. Simplified89.2%

      \[\leadsto \color{blue}{\frac{a1}{\frac{b2}{\frac{a2}{b1}}}} \]
    4. Step-by-step derivation

      1. associate-/r/90.6%

        \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    5. Applied egg-rr90.6%

      \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{a2} \cdot b1}} \]

    if -3.0000000000000001e-27 < b1

    1. Initial program 86.4%

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
    2. Step-by-step derivation

      1. times-frac86.5%

        \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

    3. Simplified86.5%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification87.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \leq -3 \cdot 10^{-27}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Alternative 6?

\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]
Derivation

  1. Initial program 87.5%

    \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
  2. Step-by-step derivation

    1. times-frac88.5%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

  3. Simplified88.5%

    \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

  4. Final simplification88.5%

    \[\leadsto \frac{a1}{b1} \cdot \frac{a2}{b2} \]

Alternative 7?

\[\frac{a2}{b2 \cdot \frac{b1}{a1}} \]
Derivation

  1. Initial program 87.5%

    \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
  2. Step-by-step derivation

    1. times-frac88.5%

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]

  3. Simplified88.5%

    \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}} \]
  4. Step-by-step derivation

    1. clear-num88.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{a1}}} \cdot \frac{a2}{b2} \]

    2. frac-times88.3%

      \[\leadsto \color{blue}{\frac{1 \cdot a2}{\frac{b1}{a1} \cdot b2}} \]

    3. *-un-lft-identity88.3%

      \[\leadsto \frac{\color{blue}{a2}}{\frac{b1}{a1} \cdot b2} \]

  5. Applied egg-rr88.3%

    \[\leadsto \color{blue}{\frac{a2}{\frac{b1}{a1} \cdot b2}} \]

  6. Final simplification88.3%

    \[\leadsto \frac{a2}{b2 \cdot \frac{b1}{a1}} \]

Reproduce

?
herbie shell --seed 2023166 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))