Average Error: 6.0 → 1.7
Time: 7.8s
Precision: binary64
\[[y, t] = \mathsf{sort}([y, t]) \\]
\[\left(x \cdot y - z \cdot y\right) \cdot t \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -4.9610827629677216 \cdot 10^{-15}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
\mathbf{if}\;y \leq -4.9610827629677216 \cdot 10^{-15}:\\
\;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\

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


\end{array}

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.9610827629677216e-15)
   (* (- x z) (* y t))
   (* t (- (* y x) (* y z)))))
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 tmp;
	if (y <= -4.9610827629677216e-15) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * ((y * x) - (y * z));
	}
	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

Original6.0
Target2.5
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.96108276296772159e-15

    1. Initial program 10.7

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

    2. Taylor expanded in y around inf 1.4

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

    if -4.96108276296772159e-15 < y

    1. Initial program 2.0

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9610827629677216 \cdot 10^{-15}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x - y \cdot z\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))