Average Error: 24.7 → 6.8
Time: 11.6s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ t_1 \leq -2.167735062580595 \cdot 10^{-297} \lor \neg \left(t_1 \leq 0\right) \end{array}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \end{array} \]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\
t_1 \leq -2.167735062580595 \cdot 10^{-297} \lor \neg \left(t_1 \leq 0\right)
\end{array}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\


\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
 (if (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
       (or (<= t_1 -2.167735062580595e-297) (not (<= t_1 0.0))))
   (fma (- y x) (- (/ z (- a t)) (/ t (- a t))) x)
   (- (+ y (+ (/ (* x z) t) (/ (* y a) t))) (+ (/ (* x a) t) (/ (* y z) t)))))
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 <= -2.167735062580595e-297) || !(t_1 <= 0.0)) {
		tmp = fma((y - x), ((z / (a - t)) - (t / (a - t))), x);
	} else {
		tmp = (y + (((x * z) / t) + ((y * a) / t))) - (((x * a) / t) + ((y * z) / t));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.7
Target9.0
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 2 regimes

  2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2.1677350625805951e-297 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

    1. Initial program 21.5

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

    2. Simplified7.4

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

    3. Applied *-un-lft-identity_binary647.4

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

    4. Applied add-cube-cbrt_binary648.1

      \[\leadsto \mathsf{fma}\left(y - x, \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{1 \cdot \left(a - t\right)}, x\right) \]

    5. Applied times-frac_binary648.1

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{1} \cdot \frac{\sqrt[3]{z - t}}{a - t}}, x\right) \]

    6. Simplified8.1

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

    7. Taylor expanded in z around 0 7.4

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

    if -2.1677350625805951e-297 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

    1. Initial program 60.6

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

    2. Simplified60.5

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

    3. Taylor expanded in t around inf 1.1

      \[\leadsto \color{blue}{\left(y + \left(\frac{z \cdot x}{t} + \frac{a \cdot y}{t}\right)\right) - \left(\frac{a \cdot x}{t} + \frac{y \cdot z}{t}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2.167735062580595 \cdot 10^{-297} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))