Average Error: 20.3 → 9.2
Time: 5.3s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.08173755248321123 \cdot 10^{302}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le -4.05570090206155 \cdot 10^{-85}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 9.23717216014945709 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 9.401606837787705 \cdot 10^{129}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]
\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}
↓
\begin{array}{l}
\mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.08173755248321123 \cdot 10^{302}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le -4.05570090206155 \cdot 10^{-85}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 9.23717216014945709 \cdot 10^{-20}:\\
\;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 9.401606837787705 \cdot 10^{129}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r750719 = x;
        double r750720 = 9.0;
        double r750721 = r750719 * r750720;
        double r750722 = y;
        double r750723 = r750721 * r750722;
        double r750724 = z;
        double r750725 = 4.0;
        double r750726 = r750724 * r750725;
        double r750727 = t;
        double r750728 = r750726 * r750727;
        double r750729 = a;
        double r750730 = r750728 * r750729;
        double r750731 = r750723 - r750730;
        double r750732 = b;
        double r750733 = r750731 + r750732;
        double r750734 = c;
        double r750735 = r750724 * r750734;
        double r750736 = r750733 / r750735;
        return r750736;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r750737 = x;
        double r750738 = 9.0;
        double r750739 = r750737 * r750738;
        double r750740 = y;
        double r750741 = r750739 * r750740;
        double r750742 = -1.0817375524832112e+302;
        bool r750743 = r750741 <= r750742;
        double r750744 = b;
        double r750745 = z;
        double r750746 = c;
        double r750747 = r750745 * r750746;
        double r750748 = r750744 / r750747;
        double r750749 = r750737 / r750745;
        double r750750 = r750746 / r750740;
        double r750751 = r750749 / r750750;
        double r750752 = r750738 * r750751;
        double r750753 = r750748 + r750752;
        double r750754 = 4.0;
        double r750755 = a;
        double r750756 = t;
        double r750757 = r750755 * r750756;
        double r750758 = r750757 / r750746;
        double r750759 = r750754 * r750758;
        double r750760 = r750753 - r750759;
        double r750761 = -4.05570090206155e-85;
        bool r750762 = r750741 <= r750761;
        double r750763 = r750737 * r750740;
        double r750764 = r750763 / r750747;
        double r750765 = r750738 * r750764;
        double r750766 = r750748 + r750765;
        double r750767 = r750756 / r750746;
        double r750768 = r750755 * r750767;
        double r750769 = r750754 * r750768;
        double r750770 = r750766 - r750769;
        double r750771 = 9.237172160149457e-20;
        bool r750772 = r750741 <= r750771;
        double r750773 = r750744 / r750745;
        double r750774 = r750773 / r750746;
        double r750775 = r750747 / r750740;
        double r750776 = r750737 / r750775;
        double r750777 = r750738 * r750776;
        double r750778 = r750774 + r750777;
        double r750779 = r750778 - r750759;
        double r750780 = 9.401606837787705e+129;
        bool r750781 = r750741 <= r750780;
        double r750782 = r750738 * r750763;
        double r750783 = r750782 / r750747;
        double r750784 = r750748 + r750783;
        double r750785 = r750784 - r750759;
        double r750786 = r750748 + r750777;
        double r750787 = r750786 - r750769;
        double r750788 = r750781 ? r750785 : r750787;
        double r750789 = r750772 ? r750779 : r750788;
        double r750790 = r750762 ? r750770 : r750789;
        double r750791 = r750743 ? r750760 : r750790;
        return r750791;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target14.4
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if (* (* x 9.0) y) < -1.0817375524832112e+302

    1. Initial program 61.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Taylor expanded around 0 59.3

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied associate-/l*20.4

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity20.4

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{\color{blue}{1 \cdot y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    7. Applied times-frac8.3

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    8. Applied associate-/r*9.9

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{\frac{x}{\frac{z}{1}}}{\frac{c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    9. Simplified9.9

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{\color{blue}{\frac{x}{z}}}{\frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    if -1.0817375524832112e+302 < (* (* x 9.0) y) < -4.05570090206155e-85

    1. Initial program 18.2

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Taylor expanded around 0 9.2

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity9.2

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{\color{blue}{1 \cdot c}}\]

    5. Applied times-frac7.1

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{t}{c}\right)}\]

    6. Simplified7.1

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\color{blue}{a} \cdot \frac{t}{c}\right)\]

    if -4.05570090206155e-85 < (* (* x 9.0) y) < 9.237172160149457e-20

    1. Initial program 17.0

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Taylor expanded around 0 7.4

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied associate-/l*8.3

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
    5. Using strategy rm

    6. Applied associate-/r*9.7

      \[\leadsto \left(\color{blue}{\frac{\frac{b}{z}}{c}} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    if 9.237172160149457e-20 < (* (* x 9.0) y) < 9.401606837787705e+129

    1. Initial program 16.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Taylor expanded around 0 7.3

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied associate-*r/7.3

      \[\leadsto \left(\frac{b}{z \cdot c} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    if 9.401606837787705e+129 < (* (* x 9.0) y)

    1. Initial program 31.5

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Taylor expanded around 0 25.1

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied associate-/l*16.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity16.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{\color{blue}{1 \cdot c}}\]

    7. Applied times-frac13.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{t}{c}\right)}\]

    8. Simplified13.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\color{blue}{a} \cdot \frac{t}{c}\right)\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.08173755248321123 \cdot 10^{302}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le -4.05570090206155 \cdot 10^{-85}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 9.23717216014945709 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 9.401606837787705 \cdot 10^{129}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))