Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Average Error: 20.1 → 8.0

Time: 22.7s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\frac{1}{\frac{c}{\left(\frac{b}{z} + 9.0 \cdot \frac{y \cdot x}{z}\right) - a \cdot \left(4.0 \cdot t\right)}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -7.030775171884016 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.796719491101506 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{1}{c}}{\frac{1}{\frac{\left(x \cdot 9.0\right) \cdot y + b}{z} - a \cdot \left(4.0 \cdot t\right)}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 8.4886610337799 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{\left(\frac{b}{z} + 9.0 \cdot \frac{y \cdot x}{z}\right) - a \cdot \left(4.0 \cdot t\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r35554076 = x;
        double r35554077 = 9.0;
        double r35554078 = r35554076 * r35554077;
        double r35554079 = y;
        double r35554080 = r35554078 * r35554079;
        double r35554081 = z;
        double r35554082 = 4.0;
        double r35554083 = r35554081 * r35554082;
        double r35554084 = t;
        double r35554085 = r35554083 * r35554084;
        double r35554086 = a;
        double r35554087 = r35554085 * r35554086;
        double r35554088 = r35554080 - r35554087;
        double r35554089 = b;
        double r35554090 = r35554088 + r35554089;
        double r35554091 = c;
        double r35554092 = r35554081 * r35554091;
        double r35554093 = r35554090 / r35554092;
        return r35554093;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r35554094 = x;
        double r35554095 = 9.0;
        double r35554096 = r35554094 * r35554095;
        double r35554097 = y;
        double r35554098 = r35554096 * r35554097;
        double r35554099 = z;
        double r35554100 = 4.0;
        double r35554101 = r35554099 * r35554100;
        double r35554102 = t;
        double r35554103 = r35554101 * r35554102;
        double r35554104 = a;
        double r35554105 = r35554103 * r35554104;
        double r35554106 = r35554098 - r35554105;
        double r35554107 = b;
        double r35554108 = r35554106 + r35554107;
        double r35554109 = c;
        double r35554110 = r35554109 * r35554099;
        double r35554111 = r35554108 / r35554110;
        double r35554112 = -inf.0;
        bool r35554113 = r35554111 <= r35554112;
        double r35554114 = 1.0;
        double r35554115 = r35554107 / r35554099;
        double r35554116 = r35554097 * r35554094;
        double r35554117 = r35554116 / r35554099;
        double r35554118 = r35554095 * r35554117;
        double r35554119 = r35554115 + r35554118;
        double r35554120 = r35554100 * r35554102;
        double r35554121 = r35554104 * r35554120;
        double r35554122 = r35554119 - r35554121;
        double r35554123 = r35554109 / r35554122;
        double r35554124 = r35554114 / r35554123;
        double r35554125 = -7.030775171884016e+38;
        bool r35554126 = r35554111 <= r35554125;
        double r35554127 = 1.796719491101506e-67;
        bool r35554128 = r35554111 <= r35554127;
        double r35554129 = r35554114 / r35554109;
        double r35554130 = r35554098 + r35554107;
        double r35554131 = r35554130 / r35554099;
        double r35554132 = r35554131 - r35554121;
        double r35554133 = r35554114 / r35554132;
        double r35554134 = r35554129 / r35554133;
        double r35554135 = 8.4886610337799e+241;
        bool r35554136 = r35554111 <= r35554135;
        double r35554137 = r35554136 ? r35554111 : r35554124;
        double r35554138 = r35554128 ? r35554134 : r35554137;
        double r35554139 = r35554126 ? r35554111 : r35554138;
        double r35554140 = r35554113 ? r35554124 : r35554139;
        return r35554140;
}

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

Target

Original	20.1
Target	14.5
Herbie	8.0

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9.0 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9.0 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 8.4886610337799e+241 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 51.7

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 24.6

      \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9.0\right) \cdot y}{z} - \left(t \cdot 4.0\right) \cdot a}{c}}\]
   3. Using strategy rm

   4. Applied clear-num 24.7

      \[\leadsto \color{blue}{\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9.0\right) \cdot y}{z} - \left(t \cdot 4.0\right) \cdot a}}}\]

   5. Taylor expanded around 0 24.6

      \[\leadsto \frac{1}{\frac{c}{\color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} - \left(t \cdot 4.0\right) \cdot a}}\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -7.030775171884016e+38 or 1.796719491101506e-67 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 8.4886610337799e+241

   1. Initial program 0.7

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   if -7.030775171884016e+38 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.796719491101506e-67

   1. Initial program 14.7

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 1.2

      \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9.0\right) \cdot y}{z} - \left(t \cdot 4.0\right) \cdot a}{c}}\]
   3. Using strategy rm

   4. Applied clear-num 1.6

      \[\leadsto \color{blue}{\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9.0\right) \cdot y}{z} - \left(t \cdot 4.0\right) \cdot a}}}\]
   5. Using strategy rm

   6. Applied div-inv 1.7

      \[\leadsto \frac{1}{\color{blue}{c \cdot \frac{1}{\frac{b + \left(x \cdot 9.0\right) \cdot y}{z} - \left(t \cdot 4.0\right) \cdot a}}}\]

   7. Applied associate-/r* 1.3

      \[\leadsto \color{blue}{\frac{\frac{1}{c}}{\frac{1}{\frac{b + \left(x \cdot 9.0\right) \cdot y}{z} - \left(t \cdot 4.0\right) \cdot a}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\frac{1}{\frac{c}{\left(\frac{b}{z} + 9.0 \cdot \frac{y \cdot x}{z}\right) - a \cdot \left(4.0 \cdot t\right)}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -7.030775171884016 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.796719491101506 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{1}{c}}{\frac{1}{\frac{\left(x \cdot 9.0\right) \cdot y + b}{z} - a \cdot \left(4.0 \cdot t\right)}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 8.4886610337799 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{\left(\frac{b}{z} + 9.0 \cdot \frac{y \cdot x}{z}\right) - a \cdot \left(4.0 \cdot t\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))