Average Error: 6.3 → 0.7
Time: 15.3s
Precision: binary64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(x, y, z \cdot t\right)\\ \mathbf{if}\;\begin{array}{l} t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ t_2 \leq -\infty \lor \neg \left(t_2 \leq 6.307159687007492 \cdot 10^{+303}\right) \end{array}:\\ \;\;\;\;2 \cdot \left(t_1 - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 - i \cdot \mathsf{fma}\left(b \cdot c, c, a \cdot c\right)\right)\\ \end{array} \]
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}
t_1 := \mathsf{fma}\left(x, y, z \cdot t\right)\\
\mathbf{if}\;\begin{array}{l}
t_2 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\
t_2 \leq -\infty \lor \neg \left(t_2 \leq 6.307159687007492 \cdot 10^{+303}\right)
\end{array}:\\
\;\;\;\;2 \cdot \left(t_1 - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(t_1 - i \cdot \mathsf{fma}\left(b \cdot c, c, a \cdot c\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
 (let* ((t_1 (fma x y (* z t))))
   (if (let* ((t_2 (* (* c (+ a (* b c))) i)))
         (or (<= t_2 (- INFINITY)) (not (<= t_2 6.307159687007492e+303))))
     (* 2.0 (- t_1 (* c (+ (* c (* b i)) (* a i)))))
     (* 2.0 (- t_1 (* i (fma (* b c) c (* a c))))))))
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 = fma(x, y, (z * t));
	double t_2 = (c * (a + (b * c))) * i;
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 6.307159687007492e+303)) {
		tmp = 2.0 * (t_1 - (c * ((c * (b * i)) + (a * i))));
	} else {
		tmp = 2.0 * (t_1 - (i * fma((b * c), c, (a * c))));
	}
	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

Original6.3
Target1.9
Herbie0.7
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 6.3071596870074922e303 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

    1. Initial program 62.5

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

    2. Simplified62.5

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

    3. Taylor expanded in c around 0 35.0

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

    4. Simplified9.8

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

    5. Taylor expanded in c around 0 5.0

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

    if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 6.3071596870074922e303

    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. Applied fma-udef_binary640.3

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

    4. Applied distribute-rgt-in_binary640.3

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

    5. Applied fma-def_binary640.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -\infty \lor \neg \left(\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 6.307159687007492 \cdot 10^{+303}\right):\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(c \cdot \left(b \cdot i\right) + a \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - i \cdot \mathsf{fma}\left(b \cdot c, c, a \cdot c\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))