Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.1 → 7.8

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -8.40974200722042392 \cdot 10^{-79}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z}}} + \mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;a \le 3.54614619959030645 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z} + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z}}} + \mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r1331462 = x;
        double r1331463 = y;
        double r1331464 = z;
        double r1331465 = r1331463 - r1331464;
        double r1331466 = t;
        double r1331467 = r1331466 - r1331462;
        double r1331468 = r1331465 * r1331467;
        double r1331469 = a;
        double r1331470 = r1331469 - r1331464;
        double r1331471 = r1331468 / r1331470;
        double r1331472 = r1331462 + r1331471;
        return r1331472;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r1331473 = a;
        double r1331474 = -8.409742007220424e-79;
        bool r1331475 = r1331473 <= r1331474;
        double r1331476 = t;
        double r1331477 = y;
        double r1331478 = z;
        double r1331479 = r1331477 - r1331478;
        double r1331480 = r1331473 - r1331478;
        double r1331481 = cbrt(r1331480);
        double r1331482 = r1331481 * r1331481;
        double r1331483 = r1331479 / r1331482;
        double r1331484 = cbrt(r1331483);
        double r1331485 = r1331484 * r1331484;
        double r1331486 = cbrt(r1331482);
        double r1331487 = r1331485 / r1331486;
        double r1331488 = r1331476 * r1331487;
        double r1331489 = cbrt(r1331481);
        double r1331490 = r1331484 / r1331489;
        double r1331491 = r1331488 * r1331490;
        double r1331492 = x;
        double r1331493 = -r1331492;
        double r1331494 = r1331479 / r1331480;
        double r1331495 = fma(r1331493, r1331494, r1331492);
        double r1331496 = r1331491 + r1331495;
        double r1331497 = 3.5461461995903064e-159;
        bool r1331498 = r1331473 <= r1331497;
        double r1331499 = r1331476 * r1331494;
        double r1331500 = r1331492 * r1331477;
        double r1331501 = r1331500 / r1331478;
        double r1331502 = r1331499 + r1331501;
        double r1331503 = r1331489 * r1331489;
        double r1331504 = r1331485 / r1331503;
        double r1331505 = r1331476 * r1331504;
        double r1331506 = r1331505 * r1331490;
        double r1331507 = r1331506 + r1331495;
        double r1331508 = r1331498 ? r1331502 : r1331507;
        double r1331509 = r1331475 ? r1331496 : r1331508;
        return r1331509;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.1
Target	11.6
Herbie	7.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -8.409742007220424e-79

   1. Initial program 22.2

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

   2. Simplified 8.0

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

   4. Applied fma-udef 8.0

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

   6. Applied sub-neg 8.0

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

   7. Applied distribute-rgt-in 8.0

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

   8. Applied associate-+l+ 5.8

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

   9. Simplified 5.8

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

   11. Applied add-cube-cbrt 6.3

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

   12. Applied associate-/r* 6.3

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

   14. Applied add-cube-cbrt 6.3

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

   15. Applied cbrt-prod 6.4

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

   16. Applied add-cube-cbrt 6.5

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

   17. Applied times-frac 6.5

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

   18. Applied associate-*r* 6.2

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

   if -8.409742007220424e-79 < a < 3.5461461995903064e-159

   1. Initial program 29.1

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

   2. Simplified 20.2

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

   4. Applied fma-udef 20.2

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

   6. Applied sub-neg 20.2

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

   7. Applied distribute-rgt-in 20.2

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

   8. Applied associate-+l+ 11.8

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

   9. Simplified 11.8

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

   10. Taylor expanded around inf 11.5

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

   if 3.5461461995903064e-159 < a

   1. Initial program 22.4

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

   2. Simplified 9.1

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

   4. Applied fma-udef 9.2

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

   6. Applied sub-neg 9.2

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

   7. Applied distribute-rgt-in 9.2

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

   8. Applied associate-+l+ 6.2

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

   9. Simplified 6.2

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

   11. Applied add-cube-cbrt 6.7

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

   12. Applied associate-/r* 6.7

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

   14. Applied add-cube-cbrt 6.9

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

   15. Applied add-cube-cbrt 6.9

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

   16. Applied times-frac 6.9

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

   17. Applied associate-*r* 6.8

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.40974200722042392 \cdot 10^{-79}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z}}} + \mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;a \le 3.54614619959030645 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z} + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{a - z}}} + \mathsf{fma}\left(-x, \frac{y - z}{a - z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :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))))