Average Error: 20.7 → 8.3
Time: 4.4s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.1961472743277424 \cdot 10^{-28}:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + \frac{9}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;z \le 2.03165913720499016 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}\\ \mathbf{elif}\;z \le 1.02952530981005718 \cdot 10^{127}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]
\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}\;z \le -3.1961472743277424 \cdot 10^{-28}:\\
\;\;\;\;\left(\frac{\frac{b}{z}}{c} + \frac{9}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{elif}\;z \le 2.03165913720499016 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}\\

\mathbf{elif}\;z \le 1.02952530981005718 \cdot 10^{127}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\

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

\end{array}
double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((z <= -3.1961472743277424e-28)) {
		VAR = ((((b / z) / c) + (9.0 / (((z * c) / y) / x))) - (4.0 * ((a * t) / c)));
	} else {
		double VAR_1;
		if ((z <= 2.0316591372049902e-81)) {
			VAR_1 = ((((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / c) / z);
		} else {
			double VAR_2;
			if ((z <= 1.0295253098100572e+127)) {
				VAR_2 = (((b / (z * c)) + (9.0 / (((z * c) / y) / x))) - (4.0 * (a / (c / t))));
			} else {
				VAR_2 = (((b / (z * c)) + ((9.0 * (y / c)) * (x / z))) - (4.0 * ((a * t) / c)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Original20.7
Target14.5
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < -3.1961472743277424e-28

    1. Initial program 29.0

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

    2. Taylor expanded around 0 13.4

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

    4. Applied associate-/l*11.3

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

    5. Applied associate-*r/11.4

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

    7. Applied associate-/l*11.3

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

    9. Applied associate-/r*8.8

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

    if -3.1961472743277424e-28 < z < 2.0316591372049902e-81

    1. Initial program 5.9

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

    3. Applied *-commutative5.9

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

    4. Applied associate-/r*5.9

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

    if 2.0316591372049902e-81 < z < 1.0295253098100572e+127

    1. Initial program 14.3

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

    2. Taylor expanded around 0 10.0

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

    4. Applied associate-/l*8.9

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

    5. Applied associate-*r/8.9

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

    7. Applied associate-/l*8.9

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

    9. Applied associate-/l*8.6

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

    if 1.0295253098100572e+127 < z

    1. Initial program 37.2

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

    2. Taylor expanded around 0 15.3

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

    4. Applied *-commutative15.3

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

    5. Applied *-commutative15.3

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

    6. Applied times-frac11.3

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

    7. Applied associate-*r*11.3

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.1961472743277424 \cdot 10^{-28}:\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + \frac{9}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;z \le 2.03165913720499016 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}\\ \mathbf{elif}\;z \le 1.02952530981005718 \cdot 10^{127}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9}{\frac{\frac{z \cdot c}{y}}{x}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))