Average Error: 7.3 → 1.5
Time: 13.7s
Precision: binary64
\[[y, t] = \mathsf{sort}([y, t]) \\]
\[\left(x \cdot y - z \cdot y\right) \cdot t \]
↓
\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 1.0881402028126704 \cdot 10^{+307}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\ \end{array} \]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\

\mathbf{elif}\;t_1 \leq 1.0881402028126704 \cdot 10^{+307}:\\
\;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\


\end{array}

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x y) (* y z)) t)))
   (if (<= t_1 (- INFINITY))
     (* (- x z) (* y t))
     (if (<= t_1 1.0881402028126704e+307)
       (* t (* y (- x z)))
       (- (* y (* x t)) (* y (* z t)))))))
double code(double x, double y, double z, double t) {
	return ((x * y) - (z * y)) * t;
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) - (y * z)) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 1.0881402028126704e+307) {
		tmp = t * (y * (x - z));
	} else {
		tmp = (y * (x * t)) - (y * (z * t));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.3
Target3.2
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 64.0

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

    2. Simplified0.3

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

    3. Taylor expanded in y around inf 0.3

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

    if -inf.0 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 1.08814020281267044e307

    1. Initial program 1.6

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

    2. Simplified7.6

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

    3. Applied *-un-lft-identity_binary647.6

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

    4. Applied associate-*l*_binary647.6

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

    5. Simplified1.6

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

    if 1.08814020281267044e307 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

    1. Initial program 63.4

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

    2. Simplified0.4

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

    3. Applied *-un-lft-identity_binary640.4

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

    4. Applied cancel-sign-sub-inv_binary640.4

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

    5. Applied distribute-rgt-in_binary640.4

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

    6. Applied distribute-lft-in_binary640.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq 1.0881402028126704 \cdot 10^{+307}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))