Average Error: 20.6 → 10.2
Time: 5.9s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.9003598243494093 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\right)\\ \mathbf{elif}\;t \le -2.292107290191715 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\ \mathbf{elif}\;t \le 1.84617460587888084 \cdot 10^{-219}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;t \le 1.34150248531036418 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{\frac{t}{c}}{\frac{1}{a}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\ \end{array}\]
\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}\;t \le -5.9003598243494093 \cdot 10^{-28}:\\
\;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\right)\\

\mathbf{elif}\;t \le -2.292107290191715 \cdot 10^{-283}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\

\mathbf{elif}\;t \le 1.84617460587888084 \cdot 10^{-219}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\

\mathbf{elif}\;t \le 1.34150248531036418 \cdot 10^{-50}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{\frac{t}{c}}{\frac{1}{a}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\

\end{array}
double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((double) (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c))));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((t <= -5.900359824349409e-28)) {
		VAR = ((double) fma(((double) -(4.0)), ((double) (t * ((double) (a / c)))), ((double) (((double) fma(((double) (9.0 * x)), y, b)) / ((double) (z * c))))));
	} else {
		double VAR_1;
		if ((t <= -2.292107290191715e-283)) {
			VAR_1 = ((double) fma(((double) -(4.0)), ((double) (((double) (t * a)) / c)), ((double) (((double) (1.0 / z)) * ((double) (((double) fma(((double) (9.0 * x)), y, b)) / c))))));
		} else {
			double VAR_2;
			if ((t <= 1.8461746058788808e-219)) {
				VAR_2 = ((double) fma(((double) -(4.0)), ((double) (((double) (t * a)) / c)), ((double) fma(9.0, ((double) (((double) (x * y)) / ((double) (z * c)))), ((double) (b / ((double) (z * c))))))));
			} else {
				double VAR_3;
				if ((t <= 1.3415024853103642e-50)) {
					VAR_3 = ((double) fma(((double) -(4.0)), ((double) (((double) (t * a)) / c)), ((double) (((double) (1.0 / z)) * ((double) (((double) fma(((double) (9.0 * x)), y, b)) / c))))));
				} else {
					VAR_3 = ((double) fma(((double) -(4.0)), ((double) (((double) (t / c)) / ((double) (1.0 / a)))), ((double) (((double) fma(x, ((double) (9.0 * y)), b)) / ((double) (z * c))))));
				}
				VAR_2 = VAR_3;
			}
			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

Original20.6
Target14.1
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -5.900359824349409e-28

    1. Initial program 27.8

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

    2. Simplified14.4

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

    4. Applied *-un-lft-identity14.4

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

    5. Applied times-frac10.0

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

    6. Simplified10.0

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

    7. Taylor expanded around 0 9.9

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

    8. Simplified9.9

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

    if -5.900359824349409e-28 < t < -2.292107290191715e-283 or 1.8461746058788808e-219 < t < 1.3415024853103642e-50

    1. Initial program 14.4

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

    2. Simplified9.3

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

    4. Applied *-un-lft-identity9.3

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

    5. Applied times-frac10.6

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

    6. Simplified10.6

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

    if -2.292107290191715e-283 < t < 1.8461746058788808e-219

    1. Initial program 11.9

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

    2. Simplified9.9

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

    3. Taylor expanded around 0 9.7

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

    4. Simplified9.7

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

    if 1.3415024853103642e-50 < t

    1. Initial program 26.6

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

    2. Simplified13.2

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

    4. Applied associate-/l*9.7

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

    6. Applied div-inv9.7

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

    7. Applied associate-/r*10.2

      \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{\frac{t}{c}}{\frac{1}{a}}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.9003598243494093 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z \cdot c}\right)\\ \mathbf{elif}\;t \le -2.292107290191715 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\ \mathbf{elif}\;t \le 1.84617460587888084 \cdot 10^{-219}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \mathsf{fma}\left(9, \frac{x \cdot y}{z \cdot c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;t \le 1.34150248531036418 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{1}{z} \cdot \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{\frac{t}{c}}{\frac{1}{a}}, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z \cdot c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(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)))