Average Error: 24.1 → 7.6
Time: 24.3s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\left(\left(\sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) + \left(x - x \cdot \frac{y - z}{a - z}\right)\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\left(\left(\sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) + \left(x - x \cdot \frac{y - z}{a - z}\right)
double f(double x, double y, double z, double t, double a) {
        double r24996285 = x;
        double r24996286 = y;
        double r24996287 = z;
        double r24996288 = r24996286 - r24996287;
        double r24996289 = t;
        double r24996290 = r24996289 - r24996285;
        double r24996291 = r24996288 * r24996290;
        double r24996292 = a;
        double r24996293 = r24996292 - r24996287;
        double r24996294 = r24996291 / r24996293;
        double r24996295 = r24996285 + r24996294;
        return r24996295;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r24996296 = y;
        double r24996297 = z;
        double r24996298 = r24996296 - r24996297;
        double r24996299 = a;
        double r24996300 = r24996299 - r24996297;
        double r24996301 = r24996298 / r24996300;
        double r24996302 = cbrt(r24996301);
        double r24996303 = cbrt(r24996302);
        double r24996304 = r24996302 * r24996302;
        double r24996305 = cbrt(r24996304);
        double r24996306 = r24996303 * r24996305;
        double r24996307 = t;
        double r24996308 = r24996306 * r24996307;
        double r24996309 = r24996308 * r24996304;
        double r24996310 = x;
        double r24996311 = r24996310 * r24996301;
        double r24996312 = r24996310 - r24996311;
        double r24996313 = r24996309 + r24996312;
        return r24996313;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.1
Target12.1
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;z \lt -1.253613105609503593846459977496550767343 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Initial program 24.1

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

  2. Simplified11.6

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

  4. Applied fma-udef11.6

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

  6. Applied sub-neg11.6

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

  7. Applied distribute-lft-in11.6

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

  8. Applied associate-+l+7.2

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

  9. Simplified7.2

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

  11. Applied add-cube-cbrt7.5

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) \cdot \sqrt[3]{\frac{y - z}{a - z}}\right)} \cdot t + \left(x - x \cdot \frac{y - z}{a - z}\right)\]

  12. Applied associate-*l*7.5

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

  14. Applied add-cube-cbrt7.5

    \[\leadsto \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) \cdot \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) \cdot \sqrt[3]{\frac{y - z}{a - z}}}} \cdot t\right) + \left(x - x \cdot \frac{y - z}{a - z}\right)\]

  15. Applied cbrt-prod7.6

    \[\leadsto \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}}}\right)} \cdot t\right) + \left(x - x \cdot \frac{y - z}{a - z}\right)\]

  16. Final simplification7.6

    \[\leadsto \left(\left(\sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}}\right) \cdot t\right) \cdot \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{\frac{y - z}{a - z}}\right) + \left(x - x \cdot \frac{y - z}{a - z}\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))