Average Error: 25.0 → 6.8
Time: 12.5s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
↓
\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -3.793508615134499 \cdot 10^{-266}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{elif}\;t_1 \leq 8.31680306445043 \cdot 10^{+296}:\\ \;\;\;\;\left(\left(x \cdot t\right) \cdot \frac{1}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{1}{\frac{a - t}{z - t}}, x\right)\\ \end{array} \]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\
\mathbf{if}\;t_1 \leq -3.793508615134499 \cdot 10^{-266}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\

\mathbf{elif}\;t_1 \leq 0:\\
\;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\

\mathbf{elif}\;t_1 \leq 8.31680306445043 \cdot 10^{+296}:\\
\;\;\;\;\left(\left(x \cdot t\right) \cdot \frac{1}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{1}{\frac{a - t}{z - t}}, 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 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_1 -3.793508615134499e-266)
     (fma (- y x) (/ (- z t) (- a t)) x)
     (if (<= t_1 0.0)
       (-
        (+ y (+ (/ (* x z) t) (/ (* y a) t)))
        (+ (/ (* x a) t) (/ (* y z) t)))
       (if (<= t_1 8.31680306445043e+296)
         (-
          (+ (* (* x t) (/ 1.0 (- a t))) (+ x (/ (* y z) (- a t))))
          (+ (/ (* x z) (- a t)) (/ (* y t) (- a t))))
         (fma (- y x) (/ 1.0 (/ (- a t) (- z t))) 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 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -3.793508615134499e-266) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else if (t_1 <= 0.0) {
		tmp = (y + (((x * z) / t) + ((y * a) / t))) - (((x * a) / t) + ((y * z) / t));
	} else if (t_1 <= 8.31680306445043e+296) {
		tmp = (((x * t) * (1.0 / (a - t))) + (x + ((y * z) / (a - t)))) - (((x * z) / (a - t)) + ((y * t) / (a - t)));
	} else {
		tmp = fma((y - x), (1.0 / ((a - t) / (z - t))), 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
Target9.4
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -3.79350861513449922e-266

    1. Initial program 21.8

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

    2. Simplified7.6

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

    if -3.79350861513449922e-266 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

    1. Initial program 58.1

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

    2. Simplified57.9

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

    3. Taylor expanded in t around inf 3.0

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

    if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 8.3168030644504298e296

    1. Initial program 2.2

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

    2. Simplified3.0

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

    3. Taylor expanded in y around 0 1.7

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

    4. Applied div-inv_binary641.7

      \[\leadsto \left(\color{blue}{\left(t \cdot x\right) \cdot \frac{1}{a - t}} + \left(\frac{y \cdot z}{a - t} + x\right)\right) - \left(\frac{z \cdot x}{a - t} + \frac{y \cdot t}{a - t}\right) \]

    if 8.3168030644504298e296 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

    1. Initial program 62.2

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

    2. Simplified17.2

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

    3. Applied clear-num_binary6417.3

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{1}{\frac{a - t}{z - t}}}, x\right) \]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -3.793508615134499 \cdot 10^{-266}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 8.31680306445043 \cdot 10^{+296}:\\ \;\;\;\;\left(\left(x \cdot t\right) \cdot \frac{1}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{1}{\frac{a - t}{z - t}}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))