Average Error: 25.0 → 9.0
Time: 7.3s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
↓
\[\begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \mathbf{elif}\;t_1 \leq 1.1842801532370239 \cdot 10^{+274}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \end{array} \]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\

\mathbf{elif}\;t_1 \leq 1.1842801532370239 \cdot 10^{+274}:\\
\;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_1 (- INFINITY))
     (* t (- (/ y (- a z)) (/ z (- a z))))
     (if (<= t_1 1.1842801532370239e+274)
       (-
        (+ (/ (* x z) (- a z)) (+ x (/ (* y t) (- a z))))
        (+ (/ (* x y) (- a z)) (/ (* z t) (- a z))))
       (fma (- y z) (/ (- t x) (- a z)) x)))))
double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t * ((y / (a - z)) - (z / (a - z)));
	} else if (t_1 <= 1.1842801532370239e+274) {
		tmp = (((x * z) / (a - z)) + (x + ((y * t) / (a - z)))) - (((x * y) / (a - z)) + ((z * t) / (a - z)));
	} else {
		tmp = fma((y - z), ((t - x) / (a - z)), 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
Target11.7
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0

    1. Initial program 64.0

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

    2. Simplified16.7

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

    3. Taylor expanded in t around inf 26.0

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

    if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.18428015323702387e274

    1. Initial program 9.1

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

    2. Simplified13.8

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

    3. Taylor expanded in y around 0 3.7

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

    if 1.18428015323702387e274 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

    1. Initial program 58.2

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

    2. Simplified16.2

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

    3. Applied *-un-lft-identity_binary6416.2

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

    4. Applied associate-/r*_binary6416.2

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 1.1842801532370239 \cdot 10^{+274}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))