Average Error: 6.5 → 0.9
Time: 10.7s
Precision: binary64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_1 := c \cdot \left(a + b \cdot c\right)\\ t_1 \leq -\infty \lor \neg \left(t_1 \leq 1.0120235025104659 \cdot 10^{+185}\right) \end{array}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \end{array} \]
2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_1 := c \cdot \left(a + b \cdot c\right)\\
t_1 \leq -\infty \lor \neg \left(t_1 \leq 1.0120235025104659 \cdot 10^{+185}\right)
\end{array}:\\
\;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\


\end{array}

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))
↓
(FPCore (x y z t a b c i)
 :precision binary64
 (if (let* ((t_1 (* c (+ a (* b c)))))
       (or (<= t_1 (- INFINITY)) (not (<= t_1 1.0120235025104659e+185))))
   (* 2.0 (- (fma x y (* z t)) (* c (+ (* c (* b i)) (* a i)))))
   (* 2.0 (- (fma t z (* x y)) (* i (* c (fma b c a)))))))
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = c * (a + (b * c));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1.0120235025104659e+185)) {
		tmp = 2.0 * (fma(x, y, (z * t)) - (c * ((c * (b * i)) + (a * i))));
	} else {
		tmp = 2.0 * (fma(t, z, (x * y)) - (i * (c * fma(b, c, a))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Target

Original6.5
Target1.9
Herbie0.9
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -inf.0 or 1.0120235025104659e185 < (*.f64 (+.f64 a (*.f64 b c)) c)

    1. Initial program 40.7

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

    2. Simplified40.7

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]

    3. Taylor expanded in c around 0 26.9

      \[\leadsto 2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - \color{blue}{\left({c}^{2} \cdot \left(i \cdot b\right) + c \cdot \left(i \cdot a\right)\right)}\right) \]

    4. Simplified7.6

      \[\leadsto 2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - \color{blue}{c \cdot \left(i \cdot \mathsf{fma}\left(c, b, a\right)\right)}\right) \]

    5. Taylor expanded in c around 0 4.2

      \[\leadsto 2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \color{blue}{\left(c \cdot \left(i \cdot b\right) + a \cdot i\right)}\right) \]

    if -inf.0 < (*.f64 (+.f64 a (*.f64 b c)) c) < 1.0120235025104659e185

    1. Initial program 0.3

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

    2. Simplified0.3

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right)} \]

    3. Taylor expanded in x around 0 0.3

      \[\leadsto 2 \cdot \left(\color{blue}{\left(y \cdot x + t \cdot z\right)} - \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right) \]

    4. Simplified0.3

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(t, z, y \cdot x\right)} - \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot i\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot \left(a + b \cdot c\right) \leq -\infty \lor \neg \left(c \cdot \left(a + b \cdot c\right) \leq 1.0120235025104659 \cdot 10^{+185}\right):\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* 2.0 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))