Average Error: 61.9 → 0
Time: 2.3m
Precision: 64
\[\Re(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))\]
↓
\[\Re(\left(\left(\left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) \cdot \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - \frac{\sqrt{3}}{2} \cdot 2\right)\right) + \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) \cdot \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) + \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - \frac{\sqrt{3}}{2} \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) i\right) + \left(\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot 5 - \frac{4}{2}\right) + 7\right) + \mathsf{fma}\left(\frac{\sqrt{3}}{2}, 4, 5 \cdot \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right)\right) i\right)\right))\]
\Re(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))
↓
\Re(\left(\left(\left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) \cdot \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - \frac{\sqrt{3}}{2} \cdot 2\right)\right) + \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) \cdot \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) + \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - \frac{\sqrt{3}}{2} \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) i\right) + \left(\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot 5 - \frac{4}{2}\right) + 7\right) + \mathsf{fma}\left(\frac{\sqrt{3}}{2}, 4, 5 \cdot \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right)\right) i\right)\right))
double f() {
        double r5433051 = 1.0;
        double r5433052 = -r5433051;
        double r5433053 = 2.0;
        double r5433054 = r5433052 / r5433053;
        double r5433055 = 3.0;
        double r5433056 = sqrt(r5433055);
        double r5433057 = r5433056 / r5433053;
        double r5433058 = /* ERROR: no complex support in C */;
        double r5433059 = r5433058 * r5433058;
        double r5433060 = r5433059 * r5433058;
        double r5433061 = r5433060 * r5433058;
        double r5433062 = -r5433053;
        double r5433063 = 0.0;
        double r5433064 = /* ERROR: no complex support in C */;
        double r5433065 = r5433064 * r5433058;
        double r5433066 = r5433065 * r5433058;
        double r5433067 = r5433066 * r5433058;
        double r5433068 = r5433061 + r5433067;
        double r5433069 = 5.0;
        double r5433070 = /* ERROR: no complex support in C */;
        double r5433071 = r5433070 * r5433058;
        double r5433072 = r5433071 * r5433058;
        double r5433073 = r5433068 + r5433072;
        double r5433074 = 4.0;
        double r5433075 = /* ERROR: no complex support in C */;
        double r5433076 = r5433075 * r5433058;
        double r5433077 = r5433073 + r5433076;
        double r5433078 = 7.0;
        double r5433079 = /* ERROR: no complex support in C */;
        double r5433080 = r5433077 + r5433079;
        double r5433081 = /* ERROR: no complex support in C */;
        return r5433081;
}
↓
double f() {
        double r5433082 = 1.0;
        double r5433083 = 2.0;
        double r5433084 = r5433082 / r5433083;
        double r5433085 = r5433084 * r5433084;
        double r5433086 = 3.0;
        double r5433087 = sqrt(r5433086);
        double r5433088 = r5433087 / r5433083;
        double r5433089 = r5433088 * r5433088;
        double r5433090 = r5433085 - r5433089;
        double r5433091 = r5433082 + r5433090;
        double r5433092 = r5433091 * r5433090;
        double r5433093 = r5433088 / r5433083;
        double r5433094 = -r5433093;
        double r5433095 = r5433094 + r5433094;
        double r5433096 = r5433088 * r5433083;
        double r5433097 = r5433095 - r5433096;
        double r5433098 = r5433095 * r5433097;
        double r5433099 = r5433092 - r5433098;
        double r5433100 = r5433091 * r5433095;
        double r5433101 = r5433097 * r5433090;
        double r5433102 = r5433100 + r5433101;
        double r5433103 = /* ERROR: no complex support in C */;
        double r5433104 = 5.0;
        double r5433105 = r5433090 * r5433104;
        double r5433106 = 4.0;
        double r5433107 = r5433106 / r5433083;
        double r5433108 = r5433105 - r5433107;
        double r5433109 = 7.0;
        double r5433110 = r5433108 + r5433109;
        double r5433111 = r5433104 * r5433095;
        double r5433112 = fma(r5433088, r5433106, r5433111);
        double r5433113 = /* ERROR: no complex support in C */;
        double r5433114 = r5433103 + r5433113;
        double r5433115 = /* ERROR: no complex support in C */;
        return r5433115;
}

Error

Derivation

  1. Initial program 61.9

    \[\Re(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))\]

  2. Simplified0

    \[\leadsto \color{blue}{\Re(\left(\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) - \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) \cdot \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - 2 \cdot \frac{\sqrt{3}}{2}\right)\right) + \left(\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - 2 \cdot \frac{\sqrt{3}}{2}\right) + \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right)\right) i\right) + \left(\left(7 + \left(\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot 5 - \frac{4}{2}\right)\right) + \mathsf{fma}\left(\frac{\sqrt{3}}{2}, 4, \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) \cdot 5\right) i\right)\right))}\]

  3. Final simplification0

    \[\leadsto \Re(\left(\left(\left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) \cdot \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - \frac{\sqrt{3}}{2} \cdot 2\right)\right) + \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) \cdot \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) + \left(\left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right) - \frac{\sqrt{3}}{2} \cdot 2\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)\right) i\right) + \left(\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot 5 - \frac{4}{2}\right) + 7\right) + \mathsf{fma}\left(\frac{\sqrt{3}}{2}, 4, 5 \cdot \left(\left(-\frac{\frac{\sqrt{3}}{2}}{2}\right) + \left(-\frac{\frac{\sqrt{3}}{2}}{2}\right)\right)\right) i\right)\right))\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore ()
  :name "3.9.2 real part (p56)"
  (re (+.c (+.c (+.c (+.c (*.c (*.c (*.c (complex (/ (- 1) 2) (/ (sqrt 3) 2)) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (*.c (*.c (*.c (complex (- 2) 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (*.c (complex 5 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (complex 4 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (complex 7 0))))