Average Error: 6.4 → 1.7
Time: 10.1s
Precision: 64
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\]
↓
\[\mathsf{fma}\left(2, \mathsf{fma}\left(x, y, z \cdot t\right), \sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{2} \cdot \left(\left(-\mathsf{fma}\left(c, b, a\right)\right) \cdot \left(c \cdot i\right)\right)\right)\right)\right)\]
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(x, y, z \cdot t\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 r790293 = 2.0;
        double r790294 = x;
        double r790295 = y;
        double r790296 = r790294 * r790295;
        double r790297 = z;
        double r790298 = t;
        double r790299 = r790297 * r790298;
        double r790300 = r790296 + r790299;
        double r790301 = a;
        double r790302 = b;
        double r790303 = c;
        double r790304 = r790302 * r790303;
        double r790305 = r790301 + r790304;
        double r790306 = r790305 * r790303;
        double r790307 = i;
        double r790308 = r790306 * r790307;
        double r790309 = r790300 - r790308;
        double r790310 = r790293 * r790309;
        return r790310;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r790311 = 2.0;
        double r790312 = x;
        double r790313 = y;
        double r790314 = z;
        double r790315 = t;
        double r790316 = r790314 * r790315;
        double r790317 = fma(r790312, r790313, r790316);
        double r790318 = sqrt(r790311);
        double r790319 = sqrt(r790318);
        double r790320 = c;
        double r790321 = b;
        double r790322 = a;
        double r790323 = fma(r790320, r790321, r790322);
        double r790324 = -r790323;
        double r790325 = i;
        double r790326 = r790320 * r790325;
        double r790327 = r790324 * r790326;
        double r790328 = r790318 * r790327;
        double r790329 = r790319 * r790328;
        double r790330 = r790319 * r790329;
        double r790331 = fma(r790311, r790317, r790330);
        return r790331;
}

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
Target1.7
Herbie1.7
\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right)\]

Derivation

  1. Initial program 6.4

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

  2. Simplified1.7

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

  4. Applied add-sqr-sqrt1.9

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

  5. Applied associate-*l*1.8

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

  7. Applied add-sqr-sqrt1.8

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

  8. Applied sqrt-prod1.7

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

  9. Applied associate-*l*1.7

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

  10. Final simplification1.7

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

Reproduce

herbie shell --seed 2019353 +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))))