Average Error: 6.2 → 2.0
Time: 11.1s
Precision: 64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]
↓
\[\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\right)\]
2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)
↓
\mathsf{fma}\left(2, \mathsf{fma}\left(t, z, x \cdot y\right), \sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r711973 = 2.0;
        double r711974 = x;
        double r711975 = y;
        double r711976 = r711974 * r711975;
        double r711977 = z;
        double r711978 = t;
        double r711979 = r711977 * r711978;
        double r711980 = r711976 + r711979;
        double r711981 = a;
        double r711982 = b;
        double r711983 = c;
        double r711984 = r711982 * r711983;
        double r711985 = r711981 + r711984;
        double r711986 = r711985 * r711983;
        double r711987 = i;
        double r711988 = r711986 * r711987;
        double r711989 = r711980 - r711988;
        double r711990 = r711973 * r711989;
        return r711990;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r711991 = 2.0;
        double r711992 = t;
        double r711993 = z;
        double r711994 = x;
        double r711995 = y;
        double r711996 = r711994 * r711995;
        double r711997 = fma(r711992, r711993, r711996);
        double r711998 = sqrt(r711991);
        double r711999 = sqrt(r711998);
        double r712000 = c;
        double r712001 = b;
        double r712002 = a;
        double r712003 = fma(r712000, r712001, r712002);
        double r712004 = -r712003;
        double r712005 = i;
        double r712006 = r712000 * r712005;
        double r712007 = r712004 * r712006;
        double r712008 = r711998 * r712007;
        double r712009 = r711999 * r712008;
        double r712010 = r711999 * r712009;
        double r712011 = fma(r711991, r711997, r712010);
        return r712011;
}

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

Original6.2
Target1.9
Herbie2.0
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\]

Derivation

  1. Initial program 6.2

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

  2. Simplified1.9

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

  3. Taylor expanded around inf 1.9

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

  4. Simplified1.9

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

  6. Applied add-sqr-sqrt2.2

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

  7. Applied associate-*l*2.1

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

  9. Applied add-sqr-sqrt2.1

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

  10. Applied sqrt-prod1.9

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

  11. Applied associate-*l*2.0

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

  12. Final simplification2.0

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

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