Average Error: 20.4 → 5.0
Time: 4.6s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -2.40571448827971033 \cdot 10^{26} \lor \neg \left(c \le 1.05955598498016459 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \frac{\mathsf{fma}\left(9, \frac{y}{\frac{c}{x}}, \frac{b}{c}\right)}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \end{array}\]
\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}\;c \le -2.40571448827971033 \cdot 10^{26} \lor \neg \left(c \le 1.05955598498016459 \cdot 10^{-12}\right):\\
\;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \frac{\mathsf{fma}\left(9, \frac{y}{\frac{c}{x}}, \frac{b}{c}\right)}{z}\right)\\

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

\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 (((c <= -2.4057144882797103e+26) || !(c <= 1.0595559849801646e-12))) {
		VAR = fma(-4.0, (a / (c / t)), (fma(9.0, (y / (c / x)), (b / c)) / z));
	} else {
		VAR = fma(-4.0, ((t * a) / c), ((fma((9.0 * x), y, b) / z) / c));
	}
	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

Original20.4
Target14.2
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -2.4057144882797103e+26 or 1.0595559849801646e-12 < c

    1. Initial program 23.4

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

    2. Simplified15.0

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

    4. Applied *-commutative15.0

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

    5. Applied associate-/l*11.6

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

    7. Applied *-commutative11.6

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

    8. Applied associate-/r*8.9

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

    9. Simplified8.9

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

    10. Taylor expanded around 0 8.7

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

    11. Simplified8.7

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

    13. Applied *-commutative8.7

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

    14. Applied associate-/l*6.1

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

    if -2.4057144882797103e+26 < c < 1.0595559849801646e-12

    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. Simplified5.2

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

    4. Applied associate-/r*2.9

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

    5. Simplified2.9

      \[\leadsto \mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\color{blue}{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}}{c}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.40571448827971033 \cdot 10^{26} \lor \neg \left(c \le 1.05955598498016459 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a}{\frac{c}{t}}, \frac{\mathsf{fma}\left(9, \frac{y}{\frac{c}{x}}, \frac{b}{c}\right)}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{t \cdot a}{c}, \frac{\frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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)))