Average Error: 1.2 → 0.4
Time: 8.0s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{0.1111111111111111 \cdot \frac{\left|{t}^{\frac{1}{3}}\right| \cdot x}{y}}{z \cdot 2} \cdot \sqrt{\sqrt[3]{t}}\right)\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{0.1111111111111111 \cdot \frac{\left|{t}^{\frac{1}{3}}\right| \cdot x}{y}}{z \cdot 2} \cdot \sqrt{\sqrt[3]{t}}\right)\right)
double f(double x, double y, double z, double t) {
        double r663133 = 1.0;
        double r663134 = 3.0;
        double r663135 = r663133 / r663134;
        double r663136 = x;
        double r663137 = y;
        double r663138 = 27.0;
        double r663139 = r663137 * r663138;
        double r663140 = r663136 / r663139;
        double r663141 = r663134 * r663140;
        double r663142 = z;
        double r663143 = 2.0;
        double r663144 = r663142 * r663143;
        double r663145 = r663141 / r663144;
        double r663146 = t;
        double r663147 = sqrt(r663146);
        double r663148 = r663145 * r663147;
        double r663149 = acos(r663148);
        double r663150 = r663135 * r663149;
        return r663150;
}
↓
double f(double x, double y, double z, double t) {
        double r663151 = 1.0;
        double r663152 = 3.0;
        double r663153 = cbrt(r663152);
        double r663154 = r663153 * r663153;
        double r663155 = r663151 / r663154;
        double r663156 = 1.0;
        double r663157 = r663156 / r663153;
        double r663158 = 0.1111111111111111;
        double r663159 = t;
        double r663160 = 0.3333333333333333;
        double r663161 = pow(r663159, r663160);
        double r663162 = fabs(r663161);
        double r663163 = x;
        double r663164 = r663162 * r663163;
        double r663165 = y;
        double r663166 = r663164 / r663165;
        double r663167 = r663158 * r663166;
        double r663168 = z;
        double r663169 = 2.0;
        double r663170 = r663168 * r663169;
        double r663171 = r663167 / r663170;
        double r663172 = cbrt(r663159);
        double r663173 = sqrt(r663172);
        double r663174 = r663171 * r663173;
        double r663175 = acos(r663174);
        double r663176 = r663157 * r663175;
        double r663177 = r663155 * r663176;
        return r663177;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.2
Target1.2
Herbie0.4
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

  1. Initial program 1.2

    \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.2

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  4. Applied *-un-lft-identity1.2

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\right)\right)\]

  9. Applied sqrt-prod0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt{\sqrt[3]{t}}\right)}\right)\right)\]

  10. Applied associate-*r*0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \color{blue}{\left(\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \sqrt{\sqrt[3]{t}}\right)}\right)\]

  11. Simplified0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \left|\sqrt[3]{t}\right|}{z \cdot 2}} \cdot \sqrt{\sqrt[3]{t}}\right)\right)\]

  12. Taylor expanded around 0 0.4

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\color{blue}{0.1111111111111111 \cdot \frac{\left|{t}^{\frac{1}{3}}\right| \cdot x}{y}}}{z \cdot 2} \cdot \sqrt{\sqrt[3]{t}}\right)\right)\]

  13. Final simplification0.4

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{0.1111111111111111 \cdot \frac{\left|{t}^{\frac{1}{3}}\right| \cdot x}{y}}{z \cdot 2} \cdot \sqrt{\sqrt[3]{t}}\right)\right)\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))