Average Error: 25.0 → 7.6
Time: 9.3s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
↓
\[\begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_2 \leq -7.184399663749514 \cdot 10^{-257}:\\ \;\;\;\;x + \left(y - x\right) \cdot t_1\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left(\frac{a \cdot \left(x \cdot z\right)}{{t}^{2}} + \left(y + \left(\frac{y \cdot a}{t} + \frac{x \cdot z}{t}\right)\right)\right) - \left(\frac{x \cdot a}{t} + \left(\frac{y \cdot z}{t} + \frac{a \cdot \left(y \cdot z\right)}{{t}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, {\left({t_1}^{-1}\right)}^{-1}, x\right)\\ \end{array} \]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
t_1 := \frac{z - t}{a - t}\\
t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\
\mathbf{if}\;t_2 \leq -7.184399663749514 \cdot 10^{-257}:\\
\;\;\;\;x + \left(y - x\right) \cdot t_1\\

\mathbf{elif}\;t_2 \leq 0:\\
\;\;\;\;\left(\frac{a \cdot \left(x \cdot z\right)}{{t}^{2}} + \left(y + \left(\frac{y \cdot a}{t} + \frac{x \cdot z}{t}\right)\right)\right) - \left(\frac{x \cdot a}{t} + \left(\frac{y \cdot z}{t} + \frac{a \cdot \left(y \cdot z\right)}{{t}^{2}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y - x, {\left({t_1}^{-1}\right)}^{-1}, x\right)\\


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -7.184399663749514e-257)
     (+ x (* (- y x) t_1))
     (if (<= t_2 0.0)
       (-
        (+ (/ (* a (* x z)) (pow t 2.0)) (+ y (+ (/ (* y a) t) (/ (* x z) t))))
        (+ (/ (* x a) t) (+ (/ (* y z) t) (/ (* a (* y z)) (pow t 2.0)))))
       (fma (- y x) (pow (pow t_1 -1.0) -1.0) x)))))
double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -7.184399663749514e-257) {
		tmp = x + ((y - x) * t_1);
	} else if (t_2 <= 0.0) {
		tmp = (((a * (x * z)) / pow(t, 2.0)) + (y + (((y * a) / t) + ((x * z) / t)))) - (((x * a) / t) + (((y * z) / t) + ((a * (y * z)) / pow(t, 2.0))));
	} else {
		tmp = fma((y - x), pow(pow(t_1, -1.0), -1.0), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original25.0
Target9.6
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -7.18439966374951385e-257

    1. Initial program 21.4

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

    2. Simplified7.2

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

    3. Applied egg-rr7.2

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

    if -7.18439966374951385e-257 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

    1. Initial program 57.7

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

    2. Simplified57.3

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

    3. Taylor expanded in a around 0 6.3

      \[\leadsto \color{blue}{\left(\frac{a \cdot \left(z \cdot x\right)}{{t}^{2}} + \left(y + \left(\frac{a \cdot y}{t} + \frac{z \cdot x}{t}\right)\right)\right) - \left(\frac{a \cdot x}{t} + \left(\frac{y \cdot z}{t} + \frac{a \cdot \left(y \cdot z\right)}{{t}^{2}}\right)\right)} \]

    if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

    1. Initial program 21.9

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

    2. Simplified8.1

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

    3. Applied egg-rr8.2

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{{\left(\frac{a - t}{z - t}\right)}^{-1}}, x\right) \]

    4. Applied egg-rr8.2

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -7.184399663749514 \cdot 10^{-257}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(\frac{a \cdot \left(x \cdot z\right)}{{t}^{2}} + \left(y + \left(\frac{y \cdot a}{t} + \frac{x \cdot z}{t}\right)\right)\right) - \left(\frac{x \cdot a}{t} + \left(\frac{y \cdot z}{t} + \frac{a \cdot \left(y \cdot z\right)}{{t}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, {\left({\left(\frac{z - t}{a - t}\right)}^{-1}\right)}^{-1}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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