Average Error: 6.4 → 1.5
Time: 10.8s
Precision: binary64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(x, y, z \cdot t\right)\\ \mathbf{if}\;i \leq -5.383784074544764 \cdot 10^{+53} \lor \neg \left(i \leq 1.0975431556761 \cdot 10^{+60}\right):\\ \;\;\;\;2 \cdot \left(t_1 - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 - c \cdot \left(i \cdot \left(a + c \cdot b\right)\right)\right)\\ \end{array} \]
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}
t_1 := \mathsf{fma}\left(x, y, z \cdot t\right)\\
\mathbf{if}\;i \leq -5.383784074544764 \cdot 10^{+53} \lor \neg \left(i \leq 1.0975431556761 \cdot 10^{+60}\right):\\
\;\;\;\;2 \cdot \left(t_1 - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(t_1 - c \cdot \left(i \cdot \left(a + c \cdot b\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
 (let* ((t_1 (fma x y (* z t))))
   (if (or (<= i -5.383784074544764e+53) (not (<= i 1.0975431556761e+60)))
     (* 2.0 (- t_1 (* i (* c (fma b c a)))))
     (* 2.0 (- t_1 (* c (* i (+ a (* c b)))))))))
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 tmp;
	if ((i <= -5.383784074544764e+53) || !(i <= 1.0975431556761e+60)) {
		tmp = 2.0 * (t_1 - (i * (c * fma(b, c, a))));
	} else {
		tmp = 2.0 * (t_1 - (c * (i * (a + (c * b)))));
	}
	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

Original6.4
Target1.8
Herbie1.5
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if i < -5.38378407454476375e53 or 1.0975431556761e60 < i

    1. Initial program 0.8

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

    2. Simplified0.7

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

    3. Applied cancel-sign-sub-inv_binary640.7

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

    if -5.38378407454476375e53 < i < 1.0975431556761e60

    1. Initial program 8.8

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

    2. Simplified8.8

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

    3. Taylor expanded in i around inf 1.9

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.383784074544764 \cdot 10^{+53} \lor \neg \left(i \leq 1.0975431556761 \cdot 10^{+60}\right):\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - i \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(x, y, z \cdot t\right) - c \cdot \left(i \cdot \left(a + c \cdot b\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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))))