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

Average Error: 24.5 → 9.4

Time: 27.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -3.400140248422175196747020283394068566805 \cdot 10^{-254}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r528500 = x;
        double r528501 = y;
        double r528502 = z;
        double r528503 = r528501 - r528502;
        double r528504 = t;
        double r528505 = r528504 - r528500;
        double r528506 = r528503 * r528505;
        double r528507 = a;
        double r528508 = r528507 - r528502;
        double r528509 = r528506 / r528508;
        double r528510 = r528500 + r528509;
        return r528510;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r528511 = x;
        double r528512 = y;
        double r528513 = z;
        double r528514 = r528512 - r528513;
        double r528515 = t;
        double r528516 = r528515 - r528511;
        double r528517 = r528514 * r528516;
        double r528518 = a;
        double r528519 = r528518 - r528513;
        double r528520 = r528517 / r528519;
        double r528521 = r528511 + r528520;
        double r528522 = -inf.0;
        bool r528523 = r528521 <= r528522;
        double r528524 = r528512 / r528513;
        double r528525 = r528511 - r528515;
        double r528526 = fma(r528524, r528525, r528515);
        double r528527 = -3.4001402484221752e-254;
        bool r528528 = r528521 <= r528527;
        double r528529 = 0.0;
        bool r528530 = r528521 <= r528529;
        double r528531 = 1.0;
        double r528532 = r528531 / r528519;
        double r528533 = r528514 * r528532;
        double r528534 = fma(r528533, r528516, r528511);
        double r528535 = r528530 ? r528526 : r528534;
        double r528536 = r528528 ? r528521 : r528535;
        double r528537 = r528523 ? r528526 : r528536;
        return r528537;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	12.2
Herbie	9.4

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -inf.0 or -3.4001402484221752e-254 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program 61.2

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

   2. Simplified 33.6

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

   4. Applied add-cube-cbrt 34.0

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

   5. Applied associate-/r* 34.0

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

   6. Taylor expanded around inf 31.9

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

   7. Simplified 23.6

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

   if -inf.0 < (+ x (/ (* (- y z) (- t x)) (- a z))) < -3.4001402484221752e-254

   1. Initial program 1.9

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

   if 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 21.5

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

   2. Simplified 7.3

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

   4. Applied div-inv 7.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -3.400140248422175196747020283394068566805 \cdot 10^{-254}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))